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Unit sum numbers of right self-injective rings

Dinesh Khurana, Ashish K. Srivastava

2007
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Bulletin of the Australian Mathematical Society
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In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to Z 2 and hence the unit sum number of a nonzero right self-injective ring is 2, w or oo. In this paper we characterise right self-injective rings with unit sum numbers w and oo. We prove that the unit sum number of a right self-injective ring R is w if and only if R has a factor ring isomorphic to Z 2 but
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... morphic to Z 2 but no factor ring isomorphic to Z2 x Z 2 , and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is 00 precisely when R has a factor ring isomorphic to Z2 x Z 2 . We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number w in which not all non-invertible elements are the sum of two units. We shall consider associative rings with identity. Our modules will be unital right modules with endomorphisms acting on the left. A ring R is said to have the n-sum property, for a positive integer n, if each of its elements can be written as a sum of exactly n units of R. It is obvious that a ring having the n-sum property also has the fc-sum property for every positive integer k > n. The unit sum number of a ring R, denoted by usn(/J), is the least integer n, if it exists, such that R has the n-sum property. If R has an element which is not a sum of units then we set usn(i?) to be 00, and if every element of R is a sum of units but R does not have the n-sum property for any n, then we set usn(/?) = u. Clearly, usn(i?) = 1 if and only if R has only one element. The unit sum number of a module M, denoted by usn(M), is the unit sum number of its endomorphism ring. The topic has been studied extensively (see [2, 3, 5, 6, 7, 10, 11, 15, 17, 19, 20] ). In 1954 Zelinsky [20] proved that usn(V D ) = 2, where V is a vector space over a division ring D, unless dim(V) = 1 and D = Z 2 , in which case usn(Vb) = 3. As End(V D ) is a (von Neumann) regular ring, in 1958 Skornyakov ([16, Problem 31, p. 167]), asked: Is every element of a regular ring a sum of units? This question was settled by

doi:10.1017/s0004972700039289
fatcat:s4bb6ts45nah7nwu63mtrp2mcu