1 Let B be a finite left regular band, k a

commutative ring with unit, and X,Y G A (B).

Foster, The idempotent elements of a

commutative ring form a Boolean Algebra, Duke math.

p] and R a

commutative ring with 1 (called also unital) of prime characteristic p with nil-radical N(R).

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any

commutative ring.

Mukesh Kumar Singh [6] analyzed a public key cryptosystem with matrices and the security of system depends on the difficulty of solving a system of multivariate congruence equations over a specified

commutative ring.

We fix a

commutative ring A and consider the category of A-weighted sets whose objects are pairs (X, f) where X is a finite set and f: X [right arrow] A is an arbitrary map called the weight of X.

Recall that the annihilator of an element r in a

commutative ring R is the collection, denoted ann(r), of elements s in R with the property that rs = 0.

Hirano, On annihilator ideals of a polynomial ring over a non

commutative ring, J.

Let K be a unital associative

commutative ring, V be a unital K-module, and [E.

In Section 2 we give the basic definitions for matroids over a

commutative ring, including representability, and we explain how they generalize the classical ones.

2](x, y)f(w) = 0 for all x, y, w [member of] R, and since R is

commutative ring with identity, so it is clear that U = where [r.

This volume contains eight chapters on the abstract algebra question of whether there is a

commutative ring with identity that has n zero-divisions.