If R is a commutative ring then the polynomial ring R[x] is semihereditary if and only if R[x] is a Bezout ring, if and only if R is von Neumann regular. A good characterization of rings such that the polynomial ring is either semihereditary or Bezout is not known in the non-commutative setting. Goursaud and Valette proved in [9] that R[x] right semihereditary implies R von Neumann regular but the converse fails to be true; for any field k, R = nโN M n (k) is a counterexample. Interesting results around this problem can be found in works by Goursaud and Valette [9], Menal [15], Moncasi and Goodearl [8], and Dicks and Schofield [5].
The characterization of semihereditary rings of power series over commutative rings was done by Brewer, Rutter and Watkins in [4]; they proved that Rโxโ is Bezout if and only if R is an โต 0 -injective von Neumann regular ring. Their motivation was to study the behavior of the weak dimension under the power series construction, characterizing nonzero Bezout power series rings as the ones of weak dimension 1. Moreover, they showed that Rโxโ is semihereditary if and only if R is an โต 0 -injective, โต 0 -complete von Neumann regular ring. These results are collected in the book by Brewer [3].
In this paper we study Bezout and semihereditary power series rings in the noncommutative setting. In Theorem 1.8 we show that the power series ring over a left โต 0 -injective von Neumann regular rings is right Bezout. We stress that this result makes a difference with the polynomial case, since it shows that for power series rings the results in the commutative case can be, at least, partially extended. The main ingredient in the proof is to find a special type of generators for principal ideals, cf. Proposition 1.5 and Corollary 1.7. In Corollary 1.9 we use this description of principal ideals to show that power series rings over non-zero left โต 0 -injective von Neumann regular rings have weak dimension 1.