• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.381-401
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• 2016

According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

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

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.1-9
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• 2006

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1511-1528
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• 2020

A skew generalized power series ring R[[S, 𝜔]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action 𝜔 of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the "untwisted" versions of all of these, and generalized power series rings. In this paper we obtain some necessary conditions on R, S and 𝜔 such that the skew generalized power series ring R[[S, 𝜔]] is (uniquely) clean. As particular cases of our general results we obtain new theorems on skew Mal'cev-Neumann series rings, skew Laurent series rings, and generalized power series rings.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.361-374
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• 2019

Let R be a ring, ($S,{\leq}$) a strictly ordered monoid, and ${\omega}:S{\rightarrow}End(R)$ a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring $R[[S,{\omega}]]$ is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and ${\omega}$ such that the skew generalized power series ring $R[[S,{\omega}]]$ is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.407-414
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• 2007

In [12], McCoy proved that if R is a commutative ring, then whenever g(x) is a zero-divisor in R[x], there exists a nonzero c $\in$ R such that cg(x) = 0. In this paper, first we extend this result to monoid rings. Then for a monoid M, we give some examples of M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings. Every reduced ring is M-quasi-Armendariz for any unique product monoid M and any strictly totally ordered monoid $(M,\;{\leq})$. Also $T_4(R)$ is M-quasi-Armendariz when R is reduced and M-Armendariz.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1067-1079
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• 2012

Let M be a right R-module, (S, ${\leq}$) a strictly totally ordered monoid which is also artinian and ${\omega}:S{\rightarrow}Aut(R)$ a monoid homomorphism, and let $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ denote the generalized inverse polynomial module over the skew generalized power series ring [[$R^{S,{\leq}},{\omega}$]]. In this paper, we prove that $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same uniform dimension as its coefficient module $M_R$, and that if, in addition, R is a right perfect ring and S is a chain monoid, then $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same couniform dimension as its coefficient module $M_R$.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.363-377
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• 2015

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.747-756
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• 1999

In ring theory it is well-known that a ring R is (von Neumann) regular if and only if all right R-modules are flat. But the analogous statement for this result does not hold for a monoid S. Hence, in sense of S-acts, Liu (]10]) showed that, as a weak analogue of this result, a monoid S is regular if and only if all left S-acts satisfying condition (E) ([6]) are flat. Moreover, Bulmann-Fleming ([6]) showed that x is a regular element of a monoid S iff the cyclic right S-act S/p(x, x2) is flat. In this paper, we show that the analogue of this result can be held for automata and them characterize regular semigroups by flat automata.