Periodic rings with commuting nilpotents
1984 ◽
Vol 7
(2)
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pp. 403-406
Keyword(s):
LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the xn=x theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.
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1961 ◽
Vol 5
(1)
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pp. 8-20
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1970 ◽
Vol 2
(1)
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pp. 107-115
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2019 ◽
Vol 56
(2)
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pp. 252-259
1987 ◽
Vol 35
(1)
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pp. 111-123
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2012 ◽
Vol 55
(2)
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pp. 418-423
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Keyword(s):
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2015 ◽
Vol DMTCS Proceedings, 27th...
(Proceedings)
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Keyword(s):
1998 ◽
Vol 40
(2)
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pp. 223-236
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