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Let A be a commutative ring, let C_r be an R-module for r=0, 1, 2, ..., and define a chain complex C__ of the form C__:...|->C_n|->C_(n-1)|->C_(n-2)|->...|->C_2|->C_1|->C_0. ...

If R is a ring (commutative with 1), the height of a prime ideal p is defined as the supremum of all n so that there is a chain p_0 subset ...p_(n-1) subset p_n=p where all ...

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence This commutative ...

A short exact sequence of groups 0-->A-->B-->C-->0 (1) is called split if it essentially presents B as the direct sum of the groups A and C. More precisely, one can construct ...

The set of all zero-systems of a group G is denoted B(G) and is called the block monoid of G since it forms a commutative monoid under the operation of zero-system addition ...

A monoid is a set that is closed under an associative binary operation and has an identity element I in S such that for all a in S, Ia=aI=a. Note that unlike a group, its ...

Also called Macaulay ring, a Cohen Macaulay ring is a Noetherian commutative unit ring R in which any proper ideal I of height n contains a sequence x_1, ..., x_n of elements ...

The homomorphism S which, according to the snake lemma, permits construction of an exact sequence (1) from the above commutative diagram with exact rows. The homomorphism S ...

An operator * for which a*b=-b*a is said to be anticommutative.

A relation R on a set S is transitive provided that for all x, y and z in S such that xRy and yRz, we also have xRz.