Primes and G-primes in \(\mathbb {Z}\)-nearalgebras

Abstract

Let N be a \(\mathbb {Z}\)-nearalgebra; that is, a left nearring with identity satisfying \( k(nn^{\prime })=(kn)n^{\prime }=n(kn^{\prime })\) for all \(k\in \mathbb {Z}\), \(n,n^{\prime }\in N\) and G be a finite group acting on N. Then the skew group nearring \(N*G\) of the group G over N is formed. If N is 3-prime (\(aNb=0\) implies \(a=0\) or \(b=0\)), then a nearring of quotients \( Q_{0}(N)\) is constructed using semigroup ideals \(A_{i}\) (a multiplicative closed set \(A_{i}\subseteq N\) such that \(A_{i}N\subseteq A_{i}\supseteq NA_{i}\)) of N and the maps \(f_{i}:A_{i}\rightarrow N\) satisfying \( (na)f_{i}=n(af_{i})\), \(n\in N\) and \(a\in A_{i}\). Through \(Q_{0}(N)\), we discuss the relationships between invariant prime subnearrings (I-primes) of \(N*G\) and G-invariant prime subnearrings (GI-primes) of N. Particularly we describe all the I-primes \(P_{i}\) of \(N*G\) such that each \( P_{i}\cap N=\{0\}\), a GI-prime of N. As an application, we settle Incomparability and Going Down Problem for N and \(N*G\) in this situation.