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If, in the above commutative diagram of modules and module homomorphisms the columns and two upper rows are exact, then so is the bottom row.

A filtration of ideals of a commutative unit ring R is a sequence of ideals ... subset= I_2 subset= I_1 subset= I_0=R, such that I_iI_j subset= I_(i+j) for all indices i,j. ...

Let K be an algebraically closed field and let I be an ideal in K(x), where x=(x_1,x_2,...,x_n) is a finite set of indeterminates. Let p in K(x) be such that for any ...

The ideal quotient (a:b) is an analog of division for ideals in a commutative ring R, (a:b)={x in R:xb subset a}. The ideal quotient is always another ideal. However, this ...

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

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

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let sigma_(ji):M_j->M_i i<=j be an R-module homomorphism. Call (M_i,sigma_(ji)) an inverse system over I ...

A polynomial map phi_(f), with f=(f_1,...,f_n) in (K[X_1,...,X_n])^m in a field K is called invertible if there exist g_1,...,g_m in K[X_1,...,x_n] such that ...

A special ideal in a commutative ring R. The Jacobson radical is the intersection of the maximal ideals in R. It could be the zero ideal, as in the case of the integers.

In a noncommutative ring R, a left ideal is a subset I which is an additive subgroup of R and such that for all r in R and all a in I, ra in I. A left ideal of R can be ...