On a Class of Regular Rings That Are Elementary Divisor Rings

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Mathematics (HMC)

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Recall that a ring R is said to be regular in the sense of yon Neumann if for every a ε R, there is an x ε R such that axa = a. If, in addition, R has an identity element 1, and there is such an x that is a unit (= element with two-sided inverse), we shall call R a unit-regular ring. This class of rings was introduced by G. Ehrlich who showed that every semi-simple Artinian ring is unit-regular, as is every strongly regular ring [2, Theorems 1 and 3]. (A ring R is called strongly regular if for every a ε R, there is an x ε R such that a2x=a. In [4, Lemma 10], it was shown that every commutative regular ring is unit-regular, and their proof carries over to the strongly regular case as is, once one observes that one-sided ideals in a strongly regular ring are two-sided. See [1].) She notes also that the ring of all linear transformations on an infinite dimensional vector space is regular, but not unitregular.

For any positive integer n, let Rn denote the ring of n X n matrices with entries in R. We call R an elementary divisor ring if for every A ε Rn, there are units P and Q in Rn such that PAQ is a diagonal matrix. (See [5].) In this note, we show that every unit-regular ring is an elementary divisor ring. From this it follows that if R is unit-regular (in particular if R is strongly regular), so is Rn. (By making use of unpublished work of Kaplansky we note that this latter result is not really new.) We show also that a necessary condition for unit-regularity given in [2, Theorem 6] is not sufficient. We close with some remarks and problems.

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© 1973 Birkhauser Verlag

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