An exact sequence is a sequence of maps(1)between a sequence of spaces , which satisfies(2)where denotes the image and the group kernel. That is, for , iff for some . It follows that . The notion of exact sequence makes sense when the spaces are groups, modules, chain complexes, or sheaves. The notation for the maps may be suppressed and the sequence written on a single line as(3)An exact sequence may be of either finite or infinite length. The special case of length five,(4)beginning and ending with zero, meaning the zero module , is called a short exact sequence. An infinite exact sequence is called a long exact sequence. For example, the sequence where and is given by multiplying by 2,(5)is a long exact sequence because at each stage the kernel and image are equal to the subgroup .Special information is conveyed when one of the spaces is the zero module. For instance, the sequence(6)is exact iff the map is injective. Similarly,(7)is exact iff the map..
The term endomorphism derives from the Greek adverb endon ("inside")and morphosis ("to form" or "to shape").In algebra, an endomorphism of a group, module, ring, vector space, etc. is a homomorphism from one object to itself (with surjectivity not required).In ergodic theory, let be a set, a sigma-algebra on and a probability measure. A map is called an endomorphism (or measure-preserving transformation) if 1. is surjective, 2. is measurable, 3. for all , where denotes the pre-image of . An endomorphism is called ergodic if it is true that implies or 1, where .
Consider a quadratic equation where and denote signed lengths. The circle which has the points and as a diameter is then called the Carlyle circle of the equation. The center of is then at the midpoint of , , which is also the midpoint of and . Call the points at which crosses the x-axis and (with ). Then(1)(2)(3)so and are the roots of the quadratic equation.
For an algebra , the associator is the trilinear map given byThe associator is identically zero iff is associative.
The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings.One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of mathematics is often referred to as abstract algebra.The word "algebra" can also refer to the "school algebra" generally taught in American middle and high schools. This includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. Mathematicians call this subject "elementary algebra," "high school algebra," "junior high..
Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. Important branches of abstract algebra are commutative algebra, representation theory, and homological algebra.Linear algebra, elementary number theory, and discrete mathematics are sometimes considered branches of abstract algebra. Ash (1998) includes the following areas in his definition of abstract algebra: logic and foundations, counting, elementary number theory, informal set theory, linear algebra, and the theory of linear operators.