Narrow Results

Let $R$ be a commutative ring with $1\ne 0$, and let $I$ be a proper ideal of $R$. Recall that $I$ is an $n$-absorbing ideal if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,\dots ,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$. We define $I$ to be a semi-$n$-absorbing ideal if $x^{n+1}\in I$ for $x\in R$ implies $x^n\in I$. More generally, for positive integers $m$ and $n$, we define $I$ to be an $(m,n)$-closed ideal if $x^m\in I$ for $x\in R$ implies $x^n\in I$. A number of examples and results on $(m,n)$-closed ideals are discussed in this paper.

For a prime ideal $P$ of a commutative ring $A$ with identity, we denote (as usual) by $O_P$ its zero-component; that is, the set of members of $P$ that are annihilated by nonmembers of $P$. We study rings in which $O_P$ is an essential ideal, whenever $P$ is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing $O_P$ with the pure part of $P$, we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.

Let $R$ be a commutative ring with $1\ne 0$. Recall that a proper ideal $I$ of $R$ is called a 2-absorbing ideal of $R$ if $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $ac\in I$ or $bc\in I$. A more general concept than 2-absorbing ideals is the concept of $n$-absorbing ideals. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal of $R$ if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1,a_2\cdots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$. The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $m$ and $n$ be integers with $1\le n<m$. A proper ideal $I$ of $R$ is called an $(m,n)$-closed ideal of $R$ if whenever $a^m\in I$ for some $a\in R$ implies $a^n\in I$. Let $A$ be a commutative ring with $1\ne 0$ and $M$ be an $A$-module. In this paper, we study $n$-absorbing ideals and $(m,n)$-closed ideals in the trivial ring extension of $A$ by $M$ (or idealization of $M$ over $A$) that is denoted by $A(+)M$.

For a commutative ring $R$ denote the Jacobson radical of the ring by $J(R)$. Khashan et al. introduced and studied J-ideals and weakly J-ideals for commutative rings with identity. In this paper, let $R$ be a non-commutative ring. We introduce weakly $\rho$-ideals for a special radical $\rho$ and show that if $\rho$ is the Jacobson radical many of the results proved by Khashan et al. are also satisfied for non-commutative rings. The results for weakly J-ideals for non-commutative rings follow as special cases of a more general situation.

Let $J(R)$ denote the Jacobson radical of a commutative ring $R$. In [H. A. Khashan and A. B. Bani-Ata, $J$-ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If $N(R)$ denotes the prime radical of a commutative ring then in [U. Tekir, S. Koc and K. H. Oral, $n$-Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an $N$ ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define $\rho$-ideals for a special radical $\rho$ and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case.