# Ring theory

## Ring theory Topics

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### Residue field

In a local ring , there is only one maximal ideal . Hence, has only one quotient ring which is a field. This field is called the residue field.

### Zero element

The identity element of an additive monoid or group or of any other algebraic structure (e.g., ring, module, abstract vector space, algebra) equipped with an addition. It is also called the additive identity and is denoted 0. The name and the symbol are borrowed from the ring of integers whose additive identity is, of course, number 0.The zero element of a ring has the property that for all and, moreover, for every element of an -module it holds that . Here, the indices distinguish the zero element of the ring from the zero element of the module. The latter also fulfils the rule for all . The notation 0 is sometimes also used for the universal bound of a Boolean algebra . In fact it behaves with respect to the operation like a zero element with respect to multiplication, since for all ...

### Cokernel

The cokernel of a group homomorphism of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space, respectively) .

### Ideal

An ideal is a subset of elements in a ring that forms an additive group and has the property that, whenever belongs to and belongs to , then and belong to . For example, the set of even integers is an ideal in the ring of integers . Given an ideal , it is possible to define a quotient ring . Ideals are commonly denoted using a Gothic typeface.A finitely generated ideal is generated by a finite list , , ..., and contains all elements of the form , where the coefficients are arbitrary elements of the ring. The list of generators is not unique, for instance in the integers.In a number ring, ideals can be represented as lattices, and can be given a finite basis of algebraic integers which generates the ideal additively. Any two bases for the same lattice are equivalent. Ideals have multiplication, and this is basically the Kronecker product of the two bases. The illustration above shows an ideal in the Gaussian integers generated by 2 and , where elements of the ideal..

### Rees ring

Given a commutative unit ring and a filtration(1)of ideals of , the Rees ring of with respect to is(2)which is the set of all formal polynomials in the variable in which the coefficient of lies in . It is a graded ring with respect to the usual addition and multiplication of polynomials, which makes it a subring of the polynomial ring . It is also a subring of the extended Rees ring(3)which is a subring of , the ring of all finite linear combinations of integer (possibly negative) powers of .If is a proper ideal of , the notation (or ) indicates the (extended) Rees ring of with respect to the -adic filtration of . If is the polynomial ring over a field , then is the coordinate ring of the blow-up of the affine space along the affine variety .

The radical of an ideal in a ring is the ideal which is the intersection of all prime ideals containing . Note that any ideal is contained in a maximal ideal, which is always prime. So the radical of an ideal is always at least as big as the original ideal. Naturally, if the ideal is prime then .Another description of the radical isThis explains the connection with the radical symbol. For example, in , consider the ideal of all polynomials with degree at least 2. Then is like a square root of . Notice that the zero set (variety) of and is the same (in because is algebraically closed). Radicals are an important part of the statement of Hilbert's Nullstellensatz.

### Zero product property

The zero product property asserts that, for elements and ,This property is especially relevant when considering algebraic structures because, e.g., integral domains are rings having the zero product property and are important objects of study because of that fact.

### Quotient ring

A quotient ring (also called a residue-class ring) is a ring that is the quotient of a ring and one of its ideals , denoted . For example, when the ring is (the integers) and the ideal is (multiples of 6), the quotient ring is .In general, a quotient ring is a set of equivalence classes where iff .The quotient ring is an integral domain iff the ideal is prime. A stronger condition occurs when the quotient ring is a field, which corresponds to when the ideal is maximal.The ideals in a quotient ring are in a one-to-one correspondence with ideals in which contain the ideal . In particular, the zero ideal in corresponds to in . In the example above from the integers, the ideal of even integers contains the ideal of the multiples of 6. In the quotient ring, the evens correspond to the ideal in ...

### Ideal quotient

The ideal quotient is an analog of division for ideals in a commutative ring ,The ideal quotient is always another ideal.However, this operation is not exactly like division. For example, when is the ring of integers, then , which is nice, while , which is not as nice.

### Ideal number

A type of number involving the roots of unity which was developed by Kummer while trying to solve Fermat's last theorem. Although factorization over the integers is unique (the fundamental theorem of algebra), factorization is not unique over the complex numbers. Over the ideal numbers, however, factorization in terms of the complex numbers becomes unique. Ideal numbers were so powerful that they were generalized by Dedekind into the more abstract ideals in general rings which are a key part of modern abstract algebra.

### Ideal extension

The extension of , an ideal in commutative ring , in a ring , is the ideal generated by its image under a ring homomorphism . Explicitly, it is any finite sum of the form where is in and is in . Sometimes the extension of is denoted .The image may not be an ideal if is not surjective. For instance, is a ring homomorphism and the image of the even integers is not an ideal since it does not contain any nonconstant polynomials. The extension of the even integers in this case is the set of polynomials with even coefficients.The extension of a prime ideal may not be prime. For example, consider . Then the extension of the even integers is not a prime ideal since .

### Proper ideal

Any ideal of a ring which is strictly smaller than the whole ring. For example, is a proper ideal of the ring of integers , since .The ideal of the polynomial ring is also proper, since it consists of all multiples of , and the constant polynomial 1 is certainly not among them.In general, an ideal of a unit ring is proper iff . The latter condition is obviously sufficient, but it is also necessary, because would imply that for all ,so that , a contradiction.Note that the above condition follows by definition: an ideal is always closed under multiplication by any element of the ring. The same property implies that an ideal containing an invertible element cannot be proper, because , where denotes the multiplicative inverse of in .Since in field all nonzero elements are invertible, it follows that the only proper ideal of is the zero ideal...

### Ideal contraction

When is a ring homomorphism and is an ideal in , then is an ideal in , called the contraction of and sometimes denoted .The contraction of a prime ideal is always prime. For example, consider . Then the contraction of is the ideal of even integers.

### Zero divisor

A nonzero element of a ring for which , where is some other nonzero element and the multiplication is the multiplication of the ring. A ring with no zero divisors is known as an integral domain. Let denote an -algebra, so that is a vector space over and(1)(2)Now define(3)where . is said to be -associative if there exists an -dimensional subspace of such that for all and . is said to be tame if is a finite union of subspaces of .The zero product property is intimately tethered to the notion of a zero divisor. For example, one may equivalently define an integral domain as a ring which satisfies the zero product property.

### Ideal class

Let be a number field with ring of integers and let be a nontrivial ideal of . Then the ideal class of , denoted , is the set of fractional ideals such that there exists a nonzero element of such that .

### Principal ring

For some authors (e.g., Bourbaki, 1964), the same as principal ideal domain. Most authors, however, do not require the ring to be an integral domain, and define a principal ring (sometimes also called a principal ideal ring) simply as a commutative unit ring (different from the zero ring) in which every ideal is principal, i.e., can be generated by a single element. Examples include the ring of integers , any field, and any polynomial ring in one variable over a field. While all Euclidean rings are principal rings, the converse is not true.If the ideal of the commutative unit ring is generated by the element of , in any quotient ring the corresponding ideal is generated by the residue class of . Hence, every quotient ring of a principal ideal ring is a principal ideal ring as well. Since is a principal ideal domain, it follows that the rings are all principal ideal rings, though not all of them are principal ideal domains.Principal ideal rings which are..

### Principal ideal domain

A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term "principal ideal domain" is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.Every Euclidean ring is a principal ideal domain, but the converse is not true. Nevertheless, the notion of greatest common divisor arising from the Euclidean algorithm can be extended to the more general context of principal ideal domains as follows. Given two nonzero elements of a principal ideal domain , a greatest common divisor of and is defined as any element of such thatEvery principal ideal domain is a unique factorization domain, but not conversely. Every polynomial ring over a field is a unique factorization domain, but it is a principal ideal domain iff the number of indeterminates is one...

### Von neumann regular ring

A von Neumann regular ring is a ring such that for all , there exists a satisfying (Jacobson 1989, p. 196).More formally, a ring is regular in the sense of von Neumann iff the following equivalent conditions hold. 1. Every -module is flat. 2. is a projective -module for every finitely generated ideal . 3. Every finitely generated right ideal is generatedby an idempotent. 4. Every finitely generated right ideal is a direct summand of .

### Principal ideal

An ideal of a ring is called principal if there is an element of such thatIn other words, the ideal is generated by the element . For example, the ideals of the ring of integers are all principal, and in fact all ideals of are principal.

### Homogeneous ideal

A homogeneous ideal in a graded ring is an ideal generated by a set of homogeneous elements, i.e., each one is contained in only one of the . For example, the polynomial ring is a graded ring, where . The ideal , i.e., all polynomials with no constant or linear terms, is a homogeneous ideal in . Another homogeneous ideal is in .Given any finite set of polynomials in variables, the process of homogenization converts them to homogeneous polynomials in variables. If is a polynomial of degree thenis the homogenization of . Similarly, if is an ideal in , then is its homogenization and is a homogeneous ideal. For example, if then . Note that in general, if then may have more elements than . However, if , ..., form a Gröbner basis using a graded monomial order, then . A polynomial is easily dehomogenized by setting the extra variable .The affine variety corresponding to a homogeneous ideal has the property that iff for all complex . Therefore, a homogeneous ideal..

### Unit ring

A unit ring is a ring with a multiplicativeidentity. It is therefore sometimes also known as a "ring with identity."It is given by a set together with two binary operators satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity: There exists an element such that for all , 4. Additive inverse: For every , there exists a such that , 5. Multiplicative associativity: For all , , 6. Multiplicative identity: There exists an element such that for all , , 7. Left and right distributivity: For all , and .

### Prime ideal

A prime ideal is an ideal such that if , then either or . For example, in the integers, the ideal (i.e., the multiples of ) is prime whenever is a prime number.In any principal ideal domain, prime ideals are generated by prime elements. Prime ideals generalize the concept of primality to more general commutative rings.An ideal is prime iff the quotient ring is an integral domain because iff . Technically, some authors choose not to allow the trivial ring as a commutative ring, in which case they usually require prime ideals to be proper ideals.A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, is a prime ideal, as it is in any integral domain. Note that this is the exception to the statement that all prime ideals in the integers are generated by prime numbers. While this might seem silly to allow this case, in some rings the structure of the prime ideals, the Zariski topology, is more interesting. For instance, in..

### Unit

A unit is an element in a ring that has a multiplicative inverse. If is an algebraic integer which divides every algebraic integer in the field, is called a unit in that field. A given field may contain an infinity of units.The units of are the elements relatively prime to . The units in which are squares are called quadratic residues.All real quadratic fields have the two units .The numbers of units in the imaginary quadratic field for , 2, ... are 4, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 6, 2, ... (OEIS A092205). There are four units for , 4, 9, 16, ... (OEIS A000290; the square numbers), six units for , 12, 27, 48, ... (OEIS A033428; three times the square numbers), and two units for all other imaginary quadratic fields, i.e., , 5, 6, 7, 8, 10, 11, ... (OEIS A092206). The following table gives the units for small . In this table, is a cube root of unity.units of 1, 23, , ..

### Prime element

A nonzero and noninvertible element of a ring which generates a prime ideal. It can also be characterized by the condition that whenever divides a product in , divides one of the factors. The prime elements of are the prime numbers .In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring , where i is the imaginary unit, is not a unique factorization domain, and there the element 2 is irreducible, but not prime, since 2 divides the product , but it does not divide any of the factors.

### Herbrand's theorem

There are two important theorems known as Herbrand's theorem.The first arises in ring theory. Let an ideal class be in if it contains an ideal whose th power is principal. Let be an odd integer and define by . Then . If and , then .The Herbrand theorem in logic states that a formula is unsatisfiable iff there is a finite set of ground clauses of that is unsatisfiable in propositional calculus. It is assumed that elements of the Herbrand base are treated as propositional variables. Since unsatisfiability is dual to validity ( is unsatisfiable iff the negation is valid), the Herbrand theorem establishes that the Herbrand universe alone is sufficient for interpretation of first-order logic. This theorem also reduces the question of unsatisfiability in first-order logic to the question of unsatisfiability in propositional calculus...

### Unique factorization domain

A unique factorization domain, called UFD for short, is any integral domain in which every nonzero noninvertible element has a unique factorization, i.e., an essentially unique decomposition as the product of prime elements or irreducible elements. In this context, the two notions coincide, since in a unique factorization domain, every irreducible element is prime, whereas the opposite implication is true in every domain.This definition arises as an application of the fundamental theorem of arithmetic, which is true in the ring of integers , to more abstract rings. Other examples of unique factorization domains are the polynomial ring , where is a field, and the ring of Gaussian integers . In general, every principal ideal domain is a unique factorization domain, but the converse is not true, since every polynomial ring is a unique factorization domain, but it is not a principal ideal domain if ...

### Primary ideal

A primary ideal is an ideal such that if , then either or for some . Prime ideals are always primary. A primary decomposition expresses any ideal as an intersection of primary ideals.

### Unique factorization

In an integral domain , the decomposition of a nonzero noninvertible element as a product of prime (or irreducible) factors(1)is unique if every other decomposition of the same type has the same number of factors(2)and its factors can be rearranged in such a way that for all indices , and differ by an invertible factor.The prime factorization of an element, if it exists, is always unique, but this does not apply, in general, to irreducible factorizations: in the ring ,(3)are two different irreducible factorizations, none of which is prime. 2 is not a prime element in , since it does not divide either of the factors of the middle expression. In fact(4)lie both outside . Furthermore,(5)which shows that is not prime either.An integral domain where every nonzero noninvertible element admits a unique irreducible factorization is called a unique factorization domain...

### Gorenstein ring

An algebraic ring which appears in treatments of duality in algebraic geometry. Let be a local Artinian ring with its maximal ideal. Then is a Gorenstein ring if the annihilator of has dimension 1 as a vector space over .

### Trivial ring

A ring defined on a singleton set . The ring operations (multiplication and addition) are defined in the only possible way,(1)and(2)It follows that this is a commutative unit ring, where is the multiplicative identity. Of course, also coincides with the additive identity, i.e., it is the so-called zero element of the ring. For this reason, the trivial ring is often denoted and also called the zero ring. In fact, the subset is the only trivial subring of the ring of integers .A unit ring is trivial whenever , since this equality implies that for all (3)A trivial ring is a trivial module over itself.

### Fractional ideal

A fractional ideal is a generalization of an ideal in a ring . Instead, a fractional ideal is contained in the number field , but has the property that there is an element such that(1)is an ideal in . In particular, every element in can be written as a fraction, with a fixed denominator.(2)Note that the multiplication of two fractional ideals is another fractional ideal.For example, in the field , the set(3)is a fractional ideal because(4)Note that , where(5)and so is an inverse to .Given any fractional ideal there is always a fractional ideal such that . Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup , and the quotient group is called the ideal class group.

### First ring isomorphism theorem

Let be a ring. If is a ring homomorphism, then is an ideal of , is a subring of , and .

### Stickelberger relation

Let be a prime ideal in not containing . Thenwhere the sum is over all which are relatively prime to . Here is the ring of integers in , , and other quantities are defined by Ireland and Rosen (1990).

### Noetherian ring

A ring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain of left (respectively, right) ideals. In this case, the ring in question is said to satisfy the ascending chain condition on left (respectively, right) ideals.A ring is said to be Noetherian if it is both left and right Noetherian. For a ring , the following are equivalent: 1. satisfies the ascending chain condition on ideals (i.e., is Noetherian). 2. Every ideal of is finitely generated. 3. Every set of ideals contains a maximalelement.

### Semiring

A semiring is a set together with two binary operators satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Multiplicative associativity: For all , , 4. Left and right distributivity: For all , and . A semiring is therefore a commutative semigroup under addition and a semigroup under multiplication. A semiring can be empty.

### Minkowski's lemma

Let be a number ring of degree with imaginary embeddings. Then every ideal class of contains an ideal such thatwhere denotes the norm of .

### Field of fractions

The ring of fractions of an integral domain. The field of fractions of the ring of integers is the rational field , and the field of fractions of the polynomial ring over a field is the field of rational functionsThe field of fractions of an integral domain is the smallest field containing , since it is obtained from by adding the least needed to make a field, namely the possibility of dividing by any nonzero element.

### Semiprime ring

Given an ideal , a semiprime ring is one for which implies for any positive . Every prime ring is semiprime.

### Maximal order

The ring of integers of a number field , denoted , is the set of algebraic integers in , which is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .

### Extension ring

A extension ring (or ring extension) of a ring is any ring of which is a subring. For example, the field of rational numbers and the ring of Gaussian integers are extension rings of the ring of integers .For every ring , the polynomial ring is a ring extension of . If is a ring extension of , and , the setis the smallest subring of containing and , and is a ring extension of . More generally, given finitely many elements of , we can considerwhich is the ring extension of in generated by .

### Semiprime ideal

A proper ideal of a ring is called semiprime if, whenever for an ideal of and some positive integer, then . In other words, the quotient ring is a semiprime ring.If is a commutative ring, this is equivalent to requiring that coincides with its radical (and in this case is also called an ideal radical). This means that, whenever a certain positive integer power of an element of belongs to , the element itself lies in . A prime ideal is certainly semiprime, but the latter is a strictly more general notion. The ideal of the ring of integers is not prime, but it is semiprime, since for all integers , is a multiple of iff is, since both 2 and 3 must appear in its prime factorization. The same argument shows that the ideal of is always semiprime if is squarefree. This is not necessarily the case when is a semiprime number, which causes a conflict in terminology.In general, the semiprime ideals of a principal ideal domain are the proper ideals whose generator has no multiple..

### Euclidean ring

A ring without zero divisors in which an integer norm and an associated division algorithm (i.e., a Euclidean algorithm) can be defined. For signed integers, the usual norm is the absolute value and the division algorithm gives the ordinary quotient and remainder. For polynomials, the norm is the degree.Important examples of Euclidean rings (besides ) are the Gaussian integers and , the ring of polynomials with complex coefficients. All Euclidean rings are also principal rings.

### Semilocal ring

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal.If is a field, the maximal ideals of the ring of polynomials in the indeterminate are the principal idealswhere is any element of . There is a one-to-one correspondence between these ideals and the elements of . Hence is semilocal if and only if is finite.A semilocal ring always has finite Krull dimension.The ring of integers is an example of a Noetherian nonsemilocal ring, since its maximal ideals are the principal ideals , where is any prime number.

### Maximal ideal

A maximal ideal of a ring is an ideal , not equal to , such that there are no ideals "in between" and . In other words, if is an ideal which contains as a subset, then either or . For example, is a maximal ideal of iff is prime, where is the ring of integers.Only in a local ring is there just one maximal ideal. For instance, in the integers, is a maximal ideal whenever is prime.A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal.

### Schur's lemma

There are at least two statements known as Schur's lemma. 1. The endomorphism ring of an irreduciblemodule is a division algebra. 2. Let , be irreducible (linear) G-spaces and a G-linear map. Then is either invertible or (Hsiang 2000, p. 3).

### Local ring

A local ring is a ring that contains a single maximal ideal. In this case, the Jacobson radical equals this maximal ideal.One property of a local ring is that the subset is precisely the set of ring units, where is the maximal ideal. This follows because, in a ring, any nonunit belongs to at least one maximal ideal.

### Dual number

A dual number is a number , where and is a matrix with the property that (such as ).

### Left ideal

In a noncommutative ring , a left ideal is a subset which is an additive subgroup of and such that for all and all ,A left ideal of can be characterized as a right ideal of the opposite ring of .In a commutative ring, the notions of right idealand left ideal coincide.

### Drinfeld ring

Let be a smooth geometrically connected projective curve over with a prime power. Let be a fixed closed point of but not necessarily -rational. A Drinfeld ring is the ring , i.e., the sections of the sheaf of regular functions over the open set . Note that the units of are the units of .As an example, let be the projective line , then .Another example is the following. Suppose that and let be given by with a separable polynomial of even positive degree and leading coefficient nonsquare in . Let the point above . Then .

### Ring spectrum

The spectrum of a ring is the set of proper primeideals,(1)The classical example is the spectrum of polynomialrings. For instance,(2)and(3)The points are, in classical algebraic geometry, algebraic varieties. Note that are maximal ideals, hence also prime.The spectrum of a ring has a topology called the Zariski topology. The closed sets are of the form(4)For example,(5)Every prime ideal is closed except for , whose closure is .

### Krull's principal ideal theorem

The most general form of this theorem states that in a commutative unit ring , the height of every proper ideal generated by elements is at most . Equality is attained if these elements form a regular sequence.Setting yields part of the original statement on principal ideals, also known under the German name Hauptidealsatz, that for every nonzero, noninvertible element of , the ideal of has height at most 1, and, moreover, iff is a non-zero divisor.It immediately follows as a corollary that every proper ideal of a Noetherian ring has finite height and that a principal ideal domain has Krull dimension equal to 1.

### Krull dimension

If is a ring (commutative with 1), the height of a prime ideal is defined as the supremum of all so that there is a chain where all are distinct prime ideals. Then, the Krull dimension of is defined as the supremum of all the heights of all its prime ideals.

### Dedekind ring

A Dedekind ring is a commutative ring in whichthe following hold. 1. It is a Noetherian ring and a integraldomain. 2. It is the set of algebraic integers in itsfield of fractions. 3. Every nonzero prime ideal is also a maximalideal. Of course, in any ring, maximal ideals are always prime. The main example of a Dedekind domain is the ring of algebraic integers in a number field, an extension field of the rational numbers. An important consequence of the above axioms is that every ideal can be written uniquely as a product of prime ideals. This compensates for the possible failure of unique factorization of elements into irreducibles.

### Ring of fractions

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,(1)(2)Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.The original ring may not embed in this ring of..

### Irreducible ring

A ring in which the zero ideal is an irreducible ideal. Every integral domain is irreducible since if and are two nonzero ideals of , and , are nonzero elements, then is a nonzero element of , which therefore cannot be the zero ideal.

### Ring homomorphism

A ring homomorphism is a map between two rings such that 1. Addition is preserved:, 2. The zero element is mapped to zero: , and 3. Multiplication is preserved: , where the operations on the left-hand side is in and on the right-hand side in . Note that a homomorphism must preserve the additive inverse map because so .A ring homomorphism for unit rings (i.e., rings with a multiplicative identity) satisfies the additional property that one multiplicative identity is mapped to the other, i.e., .

### Irreducible ideal

A proper ideal of a ring that is not the intersection of two ideals which properly contain it. In a principal ideal domain, the ideal is irreducible iff or is an irreducible element.

### Irreducible element

An element of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones (i.e., the units and the products , where is a unit). Equivalently, an element is irreducible if the only possible decompositions of into the product of two factors are of the formwhere is the multiplicative inverse of .The prime numbers and the irreducible polynomials are examples of irreducible elements. In a principal ideal domain, the irreducible elements are the generators of the nonzero prime ideals, hence the irreducible elements are exactly the prime elements. In general, however, the two notions are not equivalent.

### Coheight

The coheight of a proper ideal of a commutative Noetherian unit ring is the Krull dimension of the quotient ring .The coheight is related to the height of by the inequality(Bruns and Herzog 1998, p. 367). Equality holds for particular classes of rings, e.g., for local Cohen-Macaulay rings (Bruns and Herzog 1998, p. 58).

### Ring

A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity: There exists an element such that for all , , 4. Additive inverse: For every there exists such that , 5. Left and right distributivity: For all , and , 6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring). Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).Rings may also satisfy various optional conditions: 7. Multiplicative commutativity:..

### Right ideal

In a noncommutative ring , a right ideal is a subset which is an additive subgroup of and such that for all and all ,(1)For all , the set(2)is a right ideal of , called the right ideal generated by .In the ring of matrices with entries in , the subset(3)is a right ideal. This is evidently an additive subgroup, and the multiplication property can be easily checked,(4)It is not a left ideal, since(5)but(6)In this example, is a one-sided ideal which is not two-sided.

### Integrally closed

If and are commutative unit rings, and is a subring of , then is called integrally closed in if every element of which is integral over belongs to ; in other words, there is no proper integral extension of contained in .If is an integral domain, then is called an integrally closed domain if it is integrally closed in its field of fractions.Every unique factorization domain is an integrally closed domain; e.g., the ring of integers and every polynomial ring over a field are integrally closed domains.Being integrally closed is a local property, i.e., every localizationof an integrally closed domain is again an integrally closed domain.

### Integral element

Given a commutative unit ring and an extension ring , an element of is called integral over if it is one of the roots of a monic polynomial with coefficients in .

### Regular ring

A regular ring in the sense of commutative algebra is a commutative unit ring such that all its localizations at prime ideals are regular local rings.In contrast, a von Neumann regular ring is an object of noncommutative ring theory defined as a ring such that for all , there exists a satisfying . von Neumann regular rings are unrelated to regular rings (or regular local rings) in the sense of commutative algebra.For example, a polynomial ring over a field is always regular in the sense of commutative algebra, but is certainly not regular in the sense of von Neumann, since if is an indeterminate, then the required property is evidently not fulfilled.

### Integral closure

The integral closure of a commutative unit ring in an extension ring is the set of all elements of which are integral over . It is a subring of containing .

### Regular local ring

A regular local ring is a local ring with maximal ideal so that can be generated with exactly elements where is the Krull dimension of the ring . Equivalently, R is regular if the vector space has dimension .

### Integrable differential ideal

A differential ideal is an ideal in the ring of smooth forms on a manifold . That is, it is closed under addition, scalar multiplication, and wedge product with an arbitrary form. The ideal is called integrable if, whenever , then also , where is the exterior derivative.For example, in , the ideal(1)where the are arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form , then the ideal would not be integrable because it would not contain .Given an integral differential ideal on , a smooth map is called integrable if the pullback of every form vanishes on , i.e., . In coordinates, an integral manifold solves a system of partial differential equations. For example, using above, a map from an open set in is integral if(2)(3)(4)(5)Conversely, any system of partial differential equations can be expressed as an integrable differential ideal on a jet bundle. For instance, on corresponds to on ...

### Endomorphism ring

Given a module over a unit ring , the set of its module endomorphisms is a ring with respect to the addition of maps,and the product given by map composition,The endomorphism ring of is, in general, noncommutative, but it is always a unit ring (its unit element being the identity map on ).

### Almost unit

An almost unit is a nonunit in the integral domain of formal power series with a nonzero first coefficient, , where . Under the operation of composition, the almost units in the integral domain of formal power series over a field form a group (Henrici 1988, p. 45).

### Finite extension

An extension field is called finite if the dimension of as a vector space over (the so-called degree of over ) is finite. A finite field extension is always algebraic.Note that "finite" is a synonym for "finite-dimensional"; it does not mean "of finite cardinality" (the field of complex numbers is a finite extension, of degree 2, of the field of real numbers, but is obviously an infinite set), and it is not even equivalent to "finitely generated" (a transcendental extension is never a finite extension, but it can be generated by a single element as, for example, the field of rational functions over a field ).A ring extension is called finite if is finitely generated as a module over . An example is the ring of Gaussian integers , which is generated by as a module over . The polynomial ring , however, is not a finite ring extension of , since all systems of generators of as a -module have infinitely many elements:..