We review some methods of associating graphs to rings. The main emphasis is on zero-divisor graphs and comaximal ones. Some new results in both directions are presented. AMS Mathematics Subject Classification (2000): 13A99, 05C99, 16S50

We compare several different concepts of integer-valued polynomials on algebras and collect the few results and many open questions to be found in the literature. (2000 Math. Subj. Classification: Primary 13F20; Secondary 16S50, 13B25, 13J10, 11C08, 11C20)

We consider Mn(R), the ring of n by n matrices on the real numbers where n ≥ 2. We find all of the natural numbers that are the degree of nilpotency of some nilpotent derivations on Mn(R). Mathematics Subject Classification: 16S50, 13N15, 16N40

This article is a survey of modules whose endomorphism rings are Dedekind finite, Hopfian or co-Hopfian. We summarise the properties of such modules and present unified proofs of known results and generalisations to new structure theorems. MSC 2010. 16S50, 16D80.

In this paper, we study some properties of the exterior tensor product on the category of modules. For this, we prove that the exterior tensor product of two permutation, endo-permutation, endo-trivial and endo-monomial modules are still permutation, endo-permutation endotrivial and endo-monomial modules respectively. Also, we prove that the cap of an exterior tensor product of two modules equa...

a module m is called epi-retractable if every submodule of m is a homomorphic image of m. dually, a module m is called co-epi-retractable if it contains a copy of each of its factor modules. in special case, a ring r is called co-pli (resp. co-pri) if rr (resp. rr) is co-epi-retractable. it is proved that if r is a left principal right duo ring, then every left ideal of r is an epi-retractable ...

We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We also obtain an exact formula for the smallest number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field and as a consequenc...

An algorithm for factoring differential systems in characteristic p has been given by Cluzeau in [Cl03]. It is based on both the reduction of a matrix called p-curvature and eigenring techniques. In this paper, we generalize this algorithm to factor partial differential systems in characteristic p. We show that this factorization problem reduces effectively to the problem of simultaneous reduct...

Associating to each pre-order on the indices 1; :::;n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relation...

Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I -adic continuity of integer-valued polynomials on A. For Noetherian one-d...