• 대한수학회지
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• 제56권5호
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• pp.1403-1418
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• 2019

Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that $f(x)=c_f f_1(x)$ with $c_f{\in}R$ and $f_1(x)$ is not zero-divisor, then $c_f$ is called an annihilating content for f(x). In this case $Ann(f)=Ann(c_f )$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs ${\Gamma}(R)$ and ${\Gamma}(R[x])$ are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

• 대한수학회지
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• 제51권6호
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• pp.1251-1267
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• 2014

The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are ${\sigma}$-rigid ${\delta}$-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where ${\sigma}$ is a ring endomorphism and ${\delta}$ is a ${\sigma}$-derivation.

• East Asian mathematical journal
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• 제17권2호
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• pp.367-374
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• 2001

Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[x]-module. Park generalized Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^S]$-module, where S is a submonoid of $\mathbb{N}$($\mathbb{N}$ is the set of all natural numbers). In this paper we extend the injective property to the Laurent power series module so that if R is a ring and E is an injective left R-module, then $E[[x^{-1},x]]$ is an injective left $R[x^S]$-module.