#### Title

On a Class of Regular Rings That Are Elementary Divisor Rings

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

12-1973

#### Abstract

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 *a ^{2}x=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 *R _{n }*denote the ring of

*n X n*matrices with entries in

*R.*We call

*R*an

*elementary divisor ring*if for every

*A ε R*, there are units

_{n}*P*and

*Q*in

*R*such that

_{n }*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

*R*. (By making use of unpublished work of Kaplansky we note that this latter result is not really new.)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.

_{n}#### Rights Information

© 1973 Birkhauser Verlag

#### Terms of Use & License Information

#### DOI

10.1007/BF01228189

#### Recommended Citation

Henriksen, Melvin. 1973. On a class of regular rings that are elementary divisor rings. Archiv der Mathematik. 24(1):133-141.