TY - JOUR

T1 - Infinitely generated projective modules over pullbacks of rings

AU - Herbera, Dolors

AU - Příhoda, Pavel

PY - 2014/1/2

Y1 - 2014/1/2

N2 - We use pullbacks of rings to realize the submonoids M of (ℕ0 ∪ {∞})k, which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right R-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left R-modules is determined by the monoid D(M) defined by reversing the inequalities determining the monoid M. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters. We also provide a rich variety of examples of semilocal rings having nonfinitely generated projective modules that are finitely generated modulo the Jacobson radical. © 2013 American Mathematical Society.

AB - We use pullbacks of rings to realize the submonoids M of (ℕ0 ∪ {∞})k, which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right R-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left R-modules is determined by the monoid D(M) defined by reversing the inequalities determining the monoid M. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters. We also provide a rich variety of examples of semilocal rings having nonfinitely generated projective modules that are finitely generated modulo the Jacobson radical. © 2013 American Mathematical Society.

U2 - 10.1090/S0002-9947-2013-05798-4

DO - 10.1090/S0002-9947-2013-05798-4

M3 - Article

VL - 366

SP - 1433

EP - 1454

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -