• Title, Summary, Keyword: semisimple artinian rings

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SEMISIMPLE DIMENSION OF MODULES

  • Amirsardari, Bahram;Bagheri, Saeid
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.711-719
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    • 2018
  • In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

SEMISIMPLE ARTINIAN LOCALIZATIONS RELATED WITH V-RINGS

  • Rim, Seog-Hoon
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.839-847
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    • 1995
  • For the given torsion theory $\tau$, we study some equivalent conditions when the localized ring $R_\tau$ be semisimple artinian (Theorem 4). Using this, if $R_\tau$ is semisimple artinian ring, we study when does the given ring R become left V-ring?

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Rings Whose Simple Singular Modules are PS-Injective

  • Xiang, Yueming;Ouyang, Lunqun
    • Kyungpook Mathematical Journal
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    • v.54 no.3
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    • pp.471-476
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    • 2014
  • Let R be a ring. A right R-module M is PS-injective if every R-homomorphism $f:aR{\rightarrow}M$ for every principally small right ideal aR can be extended to $R{\rightarrow}M$. We investigate, in this paper, rings whose simple singular modules are PS-injective. New characterizations of semiprimitive rings and semisimple Artinian rings are given.

RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION

  • Han, Juncheol;Park, Sangwon
    • 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 $x,y{\in}X(R)$.

A STRUCTURE OF NONCENTRAL IDEMPOTENTS

  • Cho, Eun-Kyung;Kwak, Tai Keun;Lee, Yang;Piao, Zhelin;Seo, Yeon Sook
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.25-40
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    • 2018
  • We focus on the structure of the set of noncentral idempotents whose role is similar to one of central idempotents. We introduce the concept of quasi-Abelian rings which unit-regular rings satisfy. We first observe that the class of quasi-Abelian rings is seated between Abelian and direct finiteness. It is proved that a regular ring is directly finite if and only if it is quasi-Abelian. It is also shown that quasi-Abelian property is not left-right symmetric, but left-right symmetric when a given ring has an involution. Quasi-Abelian property is shown to do not pass to polynomial rings, comparing with Abelian property passing to polynomial rings.

MODULES WHOSE CLASSICAL PRIME SUBMODULES ARE INTERSECTIONS OF MAXIMAL SUBMODULES

  • Arabi-Kakavand, Marzieh;Behboodi, Mahmood
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.253-266
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    • 2014
  • Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

UNIT-DUO RINGS AND RELATED GRAPHS OF ZERO DIVISORS

  • Han, Juncheol;Lee, Yang;Park, Sangwon
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1629-1643
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    • 2016
  • Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called unit-duo ring if $[x]_{\ell}=[x]_r$ for all $x{\in}X$ where $[x]_{\ell}=\{ux{\mid}u{\in}G\}$ (resp. $[x]_r=\{xu{\mid}u{\in}G\}$) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted ${\tilde{\Gamma}}(R)$) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that ${\tilde{\Gamma}}(R)$ is a finite graph, R is local if and only if diam(${\tilde{\Gamma}}(R)$) = 2.