Generalizations of r-ideals of commutative rings

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 not equal 1 and L(R) be the lattice of all ideals of R. Suppose that phi:L(R) -> L(R) boolean OR {circle divide} is a function. A proper ideal I of R is called a phi-r-ideal of R if whenever ab is an element of I and Ann(a) = (0) imply that b is an element of I for each a,b is an element of R. In addition to proven many properties of phi-r-ideals, we also examine the concept of phi-r-ideals in a trivial ring extension and use them to characterize total quotient rings.