Matrix rings with summand intersection property

F. Karabacak; Adnan Tercan

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 3, page 621-626
  • ISSN: 0011-4642

Abstract

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A ring R has right SIP (SSP) if the intersection (sum) of two direct summands of R is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of R by M has SIP if and only if R has SIP and ( 1 - e ) M e = 0 for every idempotent e in R . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

How to cite

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Karabacak, F., and Tercan, Adnan. "Matrix rings with summand intersection property." Czechoslovak Mathematical Journal 53.3 (2003): 621-626. <http://eudml.org/doc/30803>.

@article{Karabacak2003,
abstract = {A ring $R$ has right SIP (SSP) if the intersection (sum) of two direct summands of $R$ is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of $R$ by $M$ has SIP if and only if $R$ has SIP and $(1-e)Me=0$ for every idempotent $e$ in $R$. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.},
author = {Karabacak, F., Tercan, Adnan},
journal = {Czechoslovak Mathematical Journal},
keywords = {modules; Summand Intersection Property; Morita invariant; direct summands; trivial extensions; triangular matrix rings; summand intersection property; summand sum property; Morita invariants},
language = {eng},
number = {3},
pages = {621-626},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Matrix rings with summand intersection property},
url = {http://eudml.org/doc/30803},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Karabacak, F.
AU - Tercan, Adnan
TI - Matrix rings with summand intersection property
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 3
SP - 621
EP - 626
AB - A ring $R$ has right SIP (SSP) if the intersection (sum) of two direct summands of $R$ is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of $R$ by $M$ has SIP if and only if $R$ has SIP and $(1-e)Me=0$ for every idempotent $e$ in $R$. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.
LA - eng
KW - modules; Summand Intersection Property; Morita invariant; direct summands; trivial extensions; triangular matrix rings; summand intersection property; summand sum property; Morita invariants
UR - http://eudml.org/doc/30803
ER -

References

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  1. Rings and Categories of Modules, Springer-Verlag, 1974. (1974) MR0417223
  2. 10.1080/00927879908826670, Comm. Algebra 27 (1999), 3875–3885. (1999) MR1699593DOI10.1080/00927879908826670
  3. 10.1080/00927878908823714, Comm. Algebra 17 (1989), 73–92. (1989) Zbl0659.16016MR0970864DOI10.1080/00927878908823714
  4. Ring Theory, Marcel Dekker, 1976. (1976) Zbl0336.16001MR0429962
  5. 10.1080/00927878908823718, Comm. Algebra 17 (1989), 135–148. (1989) Zbl0667.16020MR0970868DOI10.1080/00927878908823718
  6. Infinite Abelian Groups, University of Michigan Press, 1969. (1969) Zbl0194.04402MR0233887
  7. 10.1080/00927878608823297, Comm. Algebra 14 (1986), 21–38. (1986) Zbl0592.13008MR0814137DOI10.1080/00927878608823297

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