• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1221-1233
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• 2018

Let R be a commutative ring with non-zero identity and let NN(R) = {I | I is a nonnil ideal of R}. Let M be an R-module and let ${\phi}-tor(M)=\{x{\in}M{\mid}Ix=0\text{ for some }I{\in}NN(R)\}$. If ${\phi}or(M)=M$, then M is called a ${\phi}$-torsion module. An R-module M is said to be ${\phi}$-flat, if $0{\rightarrow}{A{\otimes}_R}\;{M{\rightarrow}B{\otimes}_R}\;{M{\rightarrow}C{\otimes}_R}\;M{\rightarrow}0$ is an exact R-sequence, for any exact sequence of R-modules $0{\rightarrow}A{\rightarrow}B{\rightarrow}C{\rightarrow}0$, where C is ${\phi}$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the ${\phi}$-flat modules over a ${\phi}-Pr{\ddot{u}}fer$ ring are investigated.

• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.219-229
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• 2013

Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.93-97
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• 1994

Previously T.Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one .tau.-closed maximal ideal (by his notation Ma $x_{\tau}$(R).neq..phi.). In this short paper we drop his overall hypothesis that Ma $x_{\tau}$(R).neq..phi. and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let .tau. be a given hereditarty torsion theory for left R-module category R-Mod. The class of all .tau.-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all .tau.-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies ([2], Proposition 1.7 and 1.10).