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Register a new user$delta$-$J$-ideals of commutative rings
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Let $mathcal{I}(R)$ be the set of all ideals of a ring $R$, $delta$ be an expansion function of $mathcal{I}(R)$. In this paper, the $delta$-$J$-ideal of a commutative ring is defined, that is, if $a, bin R$ and $abin Iin mathcal{I}(R)$, then $ain J(R)$ (the Jacobson radical of $R$) or $bin delta(I)$. Moreover, some properties of $delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
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Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c in R$ such that $0 eq abc in I$, then either $ab in I$ or $c in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 eq I_1I_2I_3 subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 subseteq I$ or $I_3subseteq I$. Among other things, it is shown that if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PIDs and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly $1$-absorbing primes.
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $Bbbk[K_{r_1,ldots,r_n}]$, where $1 leq r_1 leq cdots leq r_n$. More concretely, we prove that the class group of $Bbbk[K_{r_1,ldots,r_n}]$ is isomorphic to $mathbb{Z}^n$ if $n =3$ with $r_1 geq 2$ or $n geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $Bbbk[K_{r_1,ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals for certain $K_{r_1,ldots,r_n}$ including all cases where $Bbbk[K_{r_1,ldots,r_n}]$ is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of $Bbbk[K_{r_1,ldots,r_n}]$ in the case where it is Gorenstein.
An analog of the prime ideals for simple non-commutative rings is introduced. We prove the fundamental theorem of arithmetic for such rings. The result is used to classify the surface knots and links in the smooth 4-dimensional manifolds.
It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
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