In this paper, we consider factorizations of elements of a divisor-closed multiplicative submonoid of a ring and also factorizations of elements of a module as a product of elements coming from a divisor-closed multiplicative submonoid of the ring and another element of the module. | Turk J Math (2017) 41: 483 – 499 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Factorization with respect to a divisor-closed multiplicative submonoid of a ring 1 Ashkan NIKSERESHT1,∗, Abdulrasool AZIZI2 Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran 2 Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we consider factorizations of elements of a divisor-closed multiplicative submonoid of a ring and also factorizations of elements of a module as a product of elements coming from a divisor-closed multiplicative submonoid of the ring and another element of the module. In particular, we study uniqueness and some other properties of such factorizations and investigate the behavior of these factorizations under direct sum and product of rings and modules. Key words: Unique factorization, bounded factorization, primitive elements, pr´esimplifiability, divisor-closed multiplicative submonoid 1. Introduction Throughout this paper all rings are commutative with identity and all modules are unitary. Unless explicitly stated otherwise, we assume that all modules are nonzero. R denotes a ring and M denotes an R -module. Moreover, by U(R), J(R) , and N(R) we mean the set of units, Jacobson radical, and nilradical of R , respectively. Furthermore, Z(N ), where N ⊆ M , means the set of zero divisors of N , that is, {r ∈ R|∃0 ̸= m∈N : rm = 0}. Any other undefined notation is as in [5]. Factorization theory in commutative monoids has gained considerable attention in the last two decades, especially when the considered semigroup is the semigroup of regular elements of a commutative ring; see for example [7, 9, 10, 13–18, 21, 23]. In particular, a result of Facchini was the starting point for an entire .
