Stable Rings Generated by Their Units

We introduce the class of rings satisfying (m,1)-stable range and investigate equivalent characterizations of such rings. These give generalizations of the corresponding results by Badawi (1994), Ehrlich (1976), and Fisher and Snider (1976). 2000 Mathematics Subject Classification. 19B10, 16E50. Let R be an associative ring with identity. A ring R is said to have stable range one provided that aR+bR = R implies that a+by ∈U(R) for y ∈ R. It is well known that MR cancels from direct sums if EndRM has stable range one. For further properties of stable range one condition, we refer the reader to [1, 2, 5, 7, 9, 10, 13, 14]. Many authors have studied rings generated by their units (see [3, 4, 7, 8, 10, 12]). It was shown that every unit-regular ring in which 2 is invertible is generated by its unit (see [7, Theorem 5]) and every strongly π -regular ring in which 2 is invertible is generated by its units (see [8, Theorem 3]). So far one always investigate such rings under stable range one condition. In this paper, we generalize stable range one condition and introduce rings satisfying (m,1)-stable range so as to investigate rings generated by their units. Also we give generalizations of the corresponding results in [3, 7, 8]. Throughout, rings are associative with identity and modules are right modules. GLn(R) denotes the general linear group of R, U(R) denotes the set of units of R, and that Um(R) = {x ∈ R | ∃u1, . . . ,um ∈ U(R) such that x = u1+···+um}. Let Bij(x) = I2+xeij (i ≠ j, 1 ≤ i, j ≤ 2), [α,β] = αe11+βe22, where eij (1 ≤ i, j ≤ 2) are matrix units (1 in the i,j position and 0 elsewhere). Definition 1. The ring R is said to satisfy (m,1)-stable range provided that aR+ bR = R implies that a+by ∈U(R) for y ∈Um(R). Proposition 2. The following are equivalent: (1) The ring R satisfies (m,1)-stable range. (2) Whenever ax+b = 1, there exists y ∈Um(R) such that a+by ∈U(R). Proof. (1)⇒(2). The proof is obvious. (2)⇒(1). Given aR+bR = R, then ax+by = 1 for some x,y ∈ R. So we can find z ∈ Um(R) such that axz+b = u ∈ U(R), and then axzu−1 +bu−1 = 1. Hence we have w∈Um(R) such that a+bu−1w ∈U(R). Clearly, u−1w ∈Um(R), as desired. Proposition 3. The following are equivalent: (1) The ring R satisfies (m,1)-stable range. (2) The ring R/J(R) satisfies (m,1)-stable range.