let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...

We investigate non-unique factorization of polynomials in Zpn [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Zpn [x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Zpn [x]...

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the family, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the gen...

Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the ca...

The Schützenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Schützenberger product to the level of monoids in an algebraic category D , leading to a uniform view of the corresponding constructions for monoids (Schützenberger), ordered monoids (Pin), idempotent semirings (Klíma and Polák) and algebras over a field (Re...

We consider trace monoids i.e., free monoids where some pairs of letters are allowed to commute. We show that such monoids can be faithfully represented by 22-matrices with integer entries if and only if it they are direct products of a free commutative monoid with a free product of free commutative monoids.

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e...

The graph product is an operator mixing direct and free products. It is already known that free products and direct products of automatic monoids are automatic. The main aim of this paper is to prove that graph products of automatic monoids of finite geometric type are still automatic. A similar result for prefix-automatic monoids

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets. Most monoids considered in this paper are finite. In particular, we use the term variety of monoids for variety of finite monoids. Similarly, all languages considered in this paper are regular languages and hence their syntactic monoid is finite.