The primal element in integral domain

Abstract EN

An element x of an integral domain R is called primal if whenever x divides a product a1a2 with a1 , a2∈ R, x can be written as x = x1x2 such that xi divides ai ,i =1,2.We study whenin X2 primal in A + X B[X] or A + X B[[X]], when A ⊆ B be an extension of domains.

Also we show that if A is an integral domain and S ⊆ A a splitting multiplicative system, then A+XAS[X] is a semirigid GCD-domain if and only if A is a semirigid GCD-domain and for each two elements of S, one of them divides the other.