• East Asian mathematical journal
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• v.22 no.2
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• pp.207-213
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• 2006

Using the notion of nilpotent elements, the concept of nil subsets is introduced, and related properties are investigated. We show that a nil subset on a subalgebra (resp. (closed) ideal) is a subalgebra (resp. (closed) ideal). We also prove that in a nil algebra every ideal is a subalgebra.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.687-701
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• 2012

Let $R$ be a ring and $nil(R)$ the set of all nilpotent elements of $R$. For a subset $X$ of a ring $R$, we define $N_R(X)=\{a{\in}R{\mid}xa{\in}nil(R)$ for all $x{\in}X$}, which is called a weak annihilator of $X$ in $R$. $A$ ring $R$ is called weak zip provided that for any subset $X$ of $R$, if $N_R(Y){\subseteq}nil(R)$, then there exists a finite subset $Y{\subseteq}X$ such that $N_R(Y){\subseteq}nil(R)$, and a ring $R$ is called weak symmetric if $abc{\in}nil(R){\Rightarrow}acb{\in}nil(R)$ for all a, b, $c{\in}R$. It is shown that a generalized power series ring $[[R^{S,{\leq}}]]$ is weak zip (resp. weak symmetric) if and only if $R$ is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring $[[R^{S,{\leq}}]]$ in terms of all weak associated primes of $R$ in a very straightforward way.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.205-209
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• 1997

Hong et al. [2] and Jun et al. [4] introduced the notion of k-nil radical in a BCI-algebra, and investigated its some properties. In this paper, we discuss the further properties on the k-nil radical. Let A be a subset of a BCI-algebra X. We show that the k-nil radical of A is the union of branches. We prove that if A is an ideal then the k-nil radical [A;k] is a p-ideal of X, and that if A is a subalgebra, then the k-nil radical [A;k] is a closed p-ideal, and hence a strong ideal of X.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.945-956
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• 2020

We introduce the concept of S-exchange rings to unify various subclass of exchange rings, where S is a subset of the ring. Many properties on S-exchange rings are obtained. For instance, we show that a ring R is clean if and only if R is left U(R)-exchange, a ring R is nil clean if and only if R is left (N(R) - 1)-exchange, and that a ring R is J-clean if and only if R is left (J(R) - 1)-exchange. As a conclusion, we obtain a sufficient condition such that clean (nil clean property, respectively) can pass to corners and reprove that J-clean passes to corners by a different way.