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 kin 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}Nsubseteq A_{i}supseteq NA_{i}) of N and the maps f_{i}:A_{i}rightarrow N satisfying (na)f_{i}=n(af_{i}), nin N and ain 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.