• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1237-1246
• /
• 2022

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it a mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

• Kyungpook Mathematical Journal
• /
• v.51 no.3
• /
• pp.283-291
• /
• 2011

In this paper, a generalization of the class of semicommutative rings is investigated. A ring R is called central semicommutative if for any a, b ${\in}$ R, ab = 0 implies arb is a central element of R for each r ${\in}$ R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings.

• Kyungpook Mathematical Journal
• /
• v.58 no.3
• /
• pp.427-432
• /
• 2018

An example of a strongly central semicommutative ring which is not semicommutative is constructed. This answers a question of Bhattachafjee and Chakraborty negatively.

• Journal of the Korean Mathematical Society
• /
• v.47 no.4
• /
• pp.719-733
• /
• 2010

We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.215-223
• /
• 2007

We in this note introduce the concept of NCI rings which is a generalization of NI rings. We study the basic structure of NCI rings, concentrating rings of bounded index of nilpotency and von Neumann regular rings. We also construct suitable examples to the situations raised naturally in the process.

• Kyungpook Mathematical Journal
• /
• v.52 no.2
• /
• pp.179-188
• /
• 2012

We investigate in this paper rings containing a non-essential $nil$-injective maximal left ideal. We show that if R is a left MC2 ring containing a non-essential $nil$-injective maximal left ideal, then R is a left $nil$-injective ring. Using this result, some known results are extended.

• East Asian mathematical journal
• /
• v.17 no.2
• /
• pp.275-286
• /
• 2001

In this paper, a skew polynomial ring $R[x;\alpha]$ of a ring R with a monomorphism $\alpha$ are investigated as follows: For a reduced ring R, assume that $\alpha(P){\subseteq}P$ for any minimal prime ideal P in R. Then (i) $R[x;\alpha]$ is a reduced ring, (ii) a ring R is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring) if and only if the skew polynomial ring $R[x;\alpha]$ is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring).

• Communications of the Korean Mathematical Society
• /
• v.8 no.3
• /
• pp.345-349
• /
• 1993

• Korean Journal of Mathematics
• /
• v.23 no.1
• /
• pp.37-46
• /
• 2015

We in this note introduce the concept of semi-IFP rings which is a generalization of IFP rings. We study the basic structure of semi-IFP rings, and construct suitable examples to the situations raised naturally in the process. We also show that the semi-IFP does not go up to polynomial rings.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.511-515
• /
• 2012

We first show that the semiprimeness, primeness, and reducedness can go up to up-monoid rings. By these results we can compute the lower nilradicals of up-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to up-monoid rings.