On zero-divisors of semimodules and semialgebras

Abstract

Abstract We prove McCoy’s property for the zero-divisors of polynomials in semirings, investigate the zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule M ⁢ [ G ] {M[G]} has very few zero-divisors if and only if the S-semimodule M does so. The concept of Auslander semimodules is introduced as well. Then we introduce Ohm–Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. Finally, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras.