A NOTE ON ESSENTIAL EXTENSIONS OF SEMI-SIMPLE MODULES
2008 ◽
Vol 07
(02)
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pp. 225-230
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Keyword(s):
In a series of recent papers, Beidar, Jain and Srivastava studied the question as to when a ring R with the property that essential extensions of semi-simple right R-modules are direct sums of quasi-injectives is right Noetherian. Beidar and Jain proved that it is, when R is commutative or right q.f.d. In this note we extend their results proving the following: A ring R with this property is right Noetherian iff for some n ∈ ℕ, R/socn(RR) has ascending chain condition on essential non-two-sided right ideals (in particular, when R/socn(RR) is right q.f.d. or commutative). Also shown is the following: A ring is a right Noetherian right V-ring iff modules with essential socle are quasi-continuous/quasi-injective.
1975 ◽
Vol 16
(1)
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pp. 32-33
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Keyword(s):
1987 ◽
Vol 42
(1)
◽
pp. 69-83
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2014 ◽
Vol 3
(2)
◽
pp. 34
Keyword(s):
1949 ◽
Vol 1
(2)
◽
pp. 125-152
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1970 ◽
Vol 22
(4)
◽
pp. 839-846
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2012 ◽
Vol 49
(3)
◽
pp. 366-389
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1983 ◽
Vol 35
(1)
◽
pp. 132-142
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1971 ◽
Vol 14
(3)
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pp. 443-444
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Keyword(s):