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

• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.499-505
• /
• 1997

This paper is a continuation of [3]. We prove that if A is quasi-associative (resp. an implicative) ideal of a BCI-algebra X then the k-nil radical of A is a quasi-associative (resp. an implicative) ideal of X. We also construct the quotient algebra $X/[Z;k]$ of a BCI-algebra X by the k-nhil radical [A;k], and show that if A and B are closed ideals of BCI-algebras X and Y respectively, then

• Journal of the Korean Institute of Intelligent Systems
• /
• v.20 no.5
• /
• pp.734-740
• /
• 2010

We define and study the fuzzy nil radical of a fuzzy ideal of a commutative $\kappa$-semiring.

• Journal of the Chungcheong Mathematical Society
• /
• v.12 no.1
• /
• pp.187-191
• /
• 1999

In this paper we will show that if [G(y), x]D(x) lies in the nil radical of A for all $x{\in}A$, then GD maps A into the radical, where D and G are derivations on a Banach algebra A.

• Communications of the Korean Mathematical Society
• /
• v.14 no.2
• /
• pp.365-371
• /
• 1999

In this paper, we prove that the range of homomorphism from a C\ulcorner-algebra A into a commutative Banach algebra B whose radical is nil contains no non-zero element of the radical of B. Using this result we show that there is no non-zero homomorphism from a C\ulcorner-algebra into a commutative radical nil Banach algebra.

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.265-272
• /
• 2000

Let D be a derivation on an Banach algebra A. Suppose that [[D(x), x], D(x)] lies in the nil radical of A for all $x{\;}{\in}{\;}A$. Then D(A) is contained in the Jacobson radical of A.

• Journal of applied mathematics & informatics
• /
• v.6 no.3
• /
• pp.937-942
• /
• 1999

In this paper we will show that if [G($\chi$),$\chi$] D($\chi$) and [D($\chi$), G($\chi$)] lie in the nil radical of A for all $\chi$$\in$A, then either D or G maps A into the radical where D and G are derivations on a Banach algebra A.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.2
• /
• pp.579-594
• /
• 2014

Let R be a ring and $Nil_*$(R) be the prime radical of R. In this paper, we say that a ring R is left $Nil_*$-coherent if $Nil_*$(R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of $Nil_*$-coherent rings are also studied in terms of N-injective modules and N-flat modules.

• Journal of the Korean Mathematical Society
• /
• v.50 no.5
• /
• pp.973-989
• /
• 2013

We observe from known results that the set of nilpotent elements in Armendariz rings has an important role. The upper nilradical coincides with the prime radical in Armendariz rings. So it can be shown that the factor ring of an Armendariz ring over its prime radical is also Armendariz, with the help of Antoine's results for nil-Armendariz rings. We study the structure of rings with such property in Armendariz rings and introduce APR as a generalization. It is shown that APR is placed between Armendariz and nil-Armendariz. It is shown that an APR ring which is not Armendariz, can always be constructed from any Armendariz ring. It is also proved that a ring R is APR if and only if so is R[$x$], and that N(R[$x$]) = N(R)[$x$] when R is APR, where R[$x$] is the polynomial ring with an indeterminate $x$ over R and N(-) denotes the set of all nilpotent elements. Several kinds of APR rings are found or constructed in the precess related to ordinary ring constructions.

• Korean Journal of Mathematics
• /
• v.21 no.1
• /
• pp.23-28
• /
• 2013

The structures of finite local rings of order ${\leq}$ 16 with nonzero Jacobson radical are investigated. The whole shape of non-commutative local rings of minimal order is completely determined up to isomorphism.