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

Abstract

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