in this paper, we introduce a new generalization of weakly prime ideals called $i$-prime. suppose $r$ is a commutative ring with identity and $i$ a fixed ideal of $r$. a proper ideal $p$ of $r$ is $i$-prime if for $a, b in r$ with $ab in p-ip$ implies either $a in p$ or $b in p$. we give some characterizations of $i$-prime ideals and study some of its properties.  moreover, we give conditions  ...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R. Recall that a prime ideal P of R is σ-divided if it is comparable (under inclusion) to every σ-stable ideal I of R. A ring R is called a σ-divided ring if every prime ideal of R is σ-divided. Also a ring R is almost σ-divided r...

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...

By using a special t-level relation U(μ, t) = {(x, y) ∈ S × S|(μ(x) ∧ μ(y)) ∨ IdS(x, y) ≥ t} based on a fuzzy ideal μ of a semigroup S, which is a congruence relation, we study the roughness of soft semigroups under this special ideal of S, such as rough soft subsemigroups, rough soft ideals and rough soft prime ideals.

A convex subnearlattice of a nearlattice S containing a fixed element n∈S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal nideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In t...

Let R be a ring and let I 6= R be an ideal of R. Then I is said to be a completely prime ideal of R if R/I is a domain and is said to be completely semiprime if R/I is a reduced ring. In this paper, we introduce a new class of rings known as completely prime ideal rings. We say that a ring R is a completely prime ideal ring (CPI-ring) if every prime ideal of R is completely prime. We say that a...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R and δ a σderivation of R. We recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-invariant and δ-invariant ideal I (i.e. σ(I) ⊆ I and δ(I) ⊆ I) of R. A ring R is called a δ-divided ring...

The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$,  we mean an ideal $I$ of $R$ such that $Ineq R$.  We say that a proper ideal $I$ of a ring $R$ is a  maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$,...