On unit-regular ideals

In this paper we introduce the notion of unit-regular ideals for unital rings, which is a natural generalization of unit-regular rings. It is shown that every square matrix over unit-regular ideals admits a diagonal reduction. We also prove that a regular ideal of a unital ring is unit-regular if and only if pseudo-similarity via the ideal is similarity. Let I be an ideal of a unital ring R. We say that I is regular in case for every x ∈ I there exists y ∈ I such that x = xyx. Following Goodearl [7], a unital ring R is unit-regular provided that for every x ∈ R there exists u ∈ U(R) such that x = xux. Unit-regular rings play an important role in the structure theory of regular rings. In this paper we introduce the notion of unit-regular ideals for unital rings, which is a natural generalization of unit-regular rings. We say that an ideal I of a unital ring R is unit-regular in case for every x ∈ I, there exists u ∈ U(R) such that x = xux. Let D be a division ring, V a countably generated infinite dimensional vector space over D. Let I = {x ∈ EndDV | dimD(xV ) < ∞}. Clearly, I is an ideal of EndDV . Given any x ∈ I, we have right D-module split exact sequences 0 → Kerx → V → xV → 0 and 0 → xV → V → V/xV → 0. Then V ∼= xV ⊕ Kerx ∼= V/xV ⊕ xV ; hence, dimD(Kerx) = dimD(V/xV ) = ∞ because dimD(xV ) < ∞. By [5, Corollary], x ∈ EndD(V ) is unit-regular. Therefore I is a unit-regular ideal of EndD(V ), while EndD(V ) is not a unit-regular ring by [5, Corollary]. This shows that the notion of unit-regular ideal is a nontrivial generalization of unit-regularity for regular rings. An m× n matrix A over a unital ring R is called to admit a diagonal reduction if there exist P ∈ GLm(R) and Q ∈ GLn(R) such that PAQ is a diagonal matrix. It is well-known that every square matrix over unit-regular rings admits a diagonal reduction by invertible matrices (cf. [9, Theorem 3]). But Henriksen’s method can not be extend to unit-regular ideals. P. Ara et al. have extended this result to separative exchange rings (cf. [1, Theorem 2.4]). Let D be a division ring, V an infinite dimensional vector space over D. Set R = EndD(V ). Then R is onesided unit-regular, so it is a separative regular ring. Given any A ∈ Mn(R), by [1, Theorem 2.5], A admits a diagonal reduction. So we can find U, V ∈ GLn(R) such that UAV = diag(r1, . . . , rn). Assume now that all ri ∈ R are idempotents. Received January 21, 2003. Mathematics Subject Classification. 16E50, 16U99.