• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.1-7
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• 2006

In this paper, we investigate bounded matrices over regular rings. We observe that every bounded matrix over a regular ring can be described by idempotent matrices and invertible matrices. Let A, $B{in}M_n(R)$ be bounded matrices over a regular ring R. We prove that $(AB)^d = U(BA)^dU^{-1}$ for some $U{\in}GL_n(R)$.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.655-663
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• 2014

Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring $M_n(D)$, $n{\geq}2$, if a, b are singular matrices of the same rank, then ${\mid}o_{\ell}(a){\mid}={\mid}o_{\ell}(b){\mid}$, where $o_{\ell}(a)$ and $o_{\ell}(b)$ are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that $X(R){\neq}{\emptyset}$, $$R{{\sim_=}}{\oplus}^m_{i=1}M_n_i(D_i)$$, with $D_i$ infinite division rings of the same cardinalities or R is isomorphic to the ring of $2{\times}2$ matrices over a finite field if and only if ${\mid}o_{\ell}(x){\mid}={\mid}o_{\ell}(y){\mid}$ for all $x,y{\in}X(R)$.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.143-153
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• 2018

In this paper, we define 0-minimal (m, n)-ideals in an LA-semigroup ${\mathcal{S}}$ and prove that if R(L) is a 0-minimal right (left) ideal of S, then either $R^mL^n=\{0\}$ or $R^mL^n$ is a 0-minimal (m, n)-ideal of S for $m,n{\geq}3$.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.183-193
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• 1995

Let N be a $II_1$ factor and G be a finite group acting outerly on N. Then the crossed product algebra $M = N \rtimes G$ is also a $II_1$ factor and $N' \cap M = CI$, i.e. N is irreducible in M. Moreover, N is regular in M, in other words, M is generated by the normalizer $N_M (N)$.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.357-370
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• 2015

In this paper, we introduce the concept of (m, n)-ideals in a non-associative ordered structure, which is called an ordered Abel-Grassmann's groupoid, by generalizing the concept of (m, n)-ideals in an ordered semigroup [14]. We also study the (m, n)-regular class of an ordered AG-groupoid in terms of (m, n)-ideals.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.221-227
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• 2002

In this paper we will show that the followings ; (1) Let R be a regular local ring of dimension n. Then $A_{n-2}$(R) = 0. (2) Let R be a regular local ring of dimension n and I be an ideal in R of height 3 such that R/I is a Gorenstein ring. Then [I] = 0 in $A_{n-3}$(R). (3) Let R = V[[ $X_1$, $X_2$, …, $X_{5}$ ]]/(p+ $X_1$$^{t1}$ + $X_2$$^{t2}$ + $X_3$$^{t3}$ + $X_4$$^2$+ $X_{5}$ $^2$/), where p $\neq$2, $t_1$, $t_2$, $t_3$ are arbitrary positive integers and V is a complete discrete valuation ring with (p) = mv. Assume that R/m is algebraically closed. Then all the Chow group for R is 0 except the last Chow group.group.oup.

• Journal of applied mathematics & informatics
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• v.42 no.4
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• pp.969-983
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• 2024

For an integer ℓ ≥ 1, let ${\bar{B}}_{\ell}(n)$ denotes the number of ℓ-regular over partition pairs of n. For certain conditions of ℓ, we study the divisibility of ${\bar{B}}_{\ell}(n)$ and arithmetic properties for ${\bar{B}}_{\ell}(n)$. We further obtain infinite family of congruences modulo 2^t satisfied by ${\bar{B}}_3(n)$ employing a result of Ono and Taguchi (2005) on nilpotency of Hecke operators.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.799-808
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• 1999

We investigate when the .product of two smooth manifolds admits a weakly Lagrangian embedding. Assume M, N are oriented smooth manifolds of dimension m and n,. respectively, which admit weakly Lagrangian immersions into $C^m$ and $C^n$. If m and n are odd, then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$ In the case when m is odd and n is even, we assume further that $\chi$(N) is an even integer. Then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$. As a corollary, we obtain the result that $S^n_1\;{\times}\;S^n_2\;{\times}\;...{\times}\;S^n_k$, $\kappa$>1, admits a weakly Lagrang.ian embedding into $C^n_1+^n_2+...+^n_k$ if and only if some ni is odd.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.809-817
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• 1998

We investigate when the product of two smooth manifolds admits a weakly Lagrangian embedding. Prove that, if $M^m$ and $N^n$ are smooth manifolds such that M admits a weakly Lagrangian embedding into ${\mathbb}C^m$ whose normal bundle has a nowhere vanishing section and N admits a weakly Lagrangian immersion into ${\mathbb}C^n$, then $M \times N$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$. As a corollary, we obtain that $S^m {\times} S^n$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$ if n=1,3. We investigate the problem of whether $S^m{\times}S^n$ in general admits a weakly Lagrangian embedding into ${\mathbb} C^{m+n}$.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.91-99
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• 2003

In this paper, we show that if the ring of a Merits context (A, B, M, N, ${\psi},\;{\phi}$) with zero pairings is a strongly $\pi$-regular ring of bounded index if and only if so are A and B. Furthermore, we extend this result to the ring of a Merits context over quasi-duo strongly $\pi$-regular rings.