Generalizations of r-ideals of commutative rings

Abstract

In this study, we present generalizations of the concept of r-ideals in commutative rings with a nonzero identity. Let R be a commutative ring with 0 ≠ 1 and L(R) be the lattice of all ideals of R. Suppose that ϕ:L(R) → L(R) ∪ {Ø} is a function. A proper ideal I of R is called a ϕ-r-ideal of R if whenever ab ∈ I and Ann(a) = (0) imply that b ∈ I for each a,b ∈ R. In addition to proven many properties of ϕ-r-ideals, we also examine the concept of ϕ-r-ideals in a trivial ring extension and use them to characterize total quotient rings.

Subject Classification: (2010):

• 13A15
• 13A18

Keywords:

• r-ideals
• ϕ-prime ideals
• ϕ-r-ideals