Bezout and semihereditary power series rings.
Throughout the paper all rings are associative with 1. For a ring \(R\), \(R[\![x]\!]\) denotes the ring of power series on one commuting variable \(x\) and with coefficients in \(R\). A ring \(R\) is said to be left (right) \(\aleph_0\)-injective provided any homomorphism from a countably generated left (right) ideal of \(R\) into \(R\) extends to a left (right) \(R\)-module endomorphism of \(R\). An \(\aleph_0\)-injective ring is a right and left \(\aleph_0\)-injective ring. A ring \(R\) is said to be directly finite if for any pair of elements \(a\) and \(b\) in \(R\) the equality \(ab=1\) implies \(ba=1\). A von Neumann regular ring \(R\) is said to be left upper \(\aleph_0\)-complete if any countable subset of the lattice \(L(_RR)\) of principal left ideals of \(R\) has a least upper bound in \(L(_RR)\). The author proves that if \(R\) is a left \(\aleph_0\)-injective von Neumann regular ring, then \(R[\![x]\!]\) is right Bezout. The following statements are shown to be equivalent for a directly finite ring \(R\): (1) \(R\) is a left \(\aleph_0\)-injective von Neumann regular ring; (2) (i) \(R[\![x]\!]\) is right Bezout; (ii) If \(I\) is a countably generated left ideal of \(R\) and \(e=e^2\in R\) is such that \(I\cap Re=0\), then the natural projection \(p\colon I\oplus Re\to Re\) is given by right multiplication by an element of \(R\). Also, a ring \(R\) is a two-sided \(\aleph_0\)-injective, left (or right) upper \(\aleph_0\)-complete von Neumann regular ring if and only if \(R\) is directly finite and \(R[\![x]\!]\) is a semihereditary Bezout ring.
scientific article; zbMATH DE number 2026920
scientific article