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- W2023621622 abstract "We construct a family of left and right perfect rings whose left and right Loewy lengths are arbitrary preassigned infinite ordinals. Let R be a left perfect ring with Jacobson radical N. Bass [21 defines a well-ordered chain of ideals { N, I of R by No = (0), Na, =the inverse image of the right socle of R/NO for a=,=l+ 1, and Na UjB<,Nl for a a limit ordinal. Since R is left perfect, there exists a least ordinal ao such that NA0o =R. ao is called the right Loewy length of R. If R is right perfect, a symmetrical definition of the left Loewy length of R holds. In the case that R is semiprimary, this is precisely the traditional definition (see for example [1, p. 104]), and the left and right Loewy lengths are equal (they are both equal to the index of nilpotency of N). Bass asks if there are any restrictions on the right Loewy length of a left perfect ring R. Since RR is finitely generated, this length cannot be a limit ordinal. We show that this is the only restriction. Indeed, let a and ,B be any pair of infinite ordinals. Then there exists a ring R which is right and left perfect and has left Loewy length a +1 and right Loewy length 3+1. The construction is a modification of an example in [2]. Let (I, -<) be any partially ordered set. Let R be the set of all I XI matrices (a,j) with coefficients in a field F such that a1,, = aj,j for all i, jEi, and if itj then aj j=0 except for a finite set of pairs (i, j) where i-<j. Then R is a ring under matrix addition and multiplication, and N= { (aij)EERI ai t=O for all iCI} is the Jacobson radical of R. PROPOSITION. If I has d.c.c. (respectively a.c.c.), then R is right (left ) perfect. PROOF. Since R/N= F, we need only show that N is right (left) T-nilpotent. Let I r, } be a sequence of elements of N. Let sn = rnrn1 . . . r1ro (tn = ror1 . . . rnirn). Let Fn (Gn) be the finite subset of I consisting of all those iCI such that there exists kEEI with Received by the editors July 17, 1970. AMS 1970 subject classifications. Primary 16A22, 16A48; Secondary 16A42." @default.
- W2023621622 created "2016-06-24" @default.
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- W2023621622 date "1971-02-01" @default.
- W2023621622 modified "2023-09-24" @default.
- W2023621622 title "Loewy length of perfect rings" @default.
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- W2023621622 doi "https://doi.org/10.1090/s0002-9939-1971-0276259-2" @default.
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