Journal of Commutative Algebra

Ideal class groups of monoid algebras

Husney Parvez Sarwar

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$ Theorem~\ref {6t2} (b), (d). In the case $M=N$, we prove the same without the assumption that the ring extension is reduced Theorem~\ref {6t2} (c), (d).

Article information

J. Commut. Algebra, Volume 9, Number 2 (2017), 303-312.

First available in Project Euclid: 3 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A15: Ideals; multiplicative ideal theory 13B99: None of the above, but in this section 13C10: Projective and free modules and ideals [See also 19A13]

Invertible modules monoid extensions monoid algebras


Sarwar, Husney Parvez. Ideal class groups of monoid algebras. J. Commut. Algebra 9 (2017), no. 2, 303--312. doi:10.1216/JCA-2017-9-2-303.

Export citation


  • D.F. Anderson, Projective modules over subrings of $k[X,Y]$ generated by monomials, Pacific J. Math. 79 (1978), 5–17.
  • ––––, Seminormal graded rings II, J. Pure Appl. Alg. 23 (1982), 221–226.
  • ––––, The Picard group of a monoid domain, J. Algebra 115 (1988), no. 2, 342-351.
  • W. Bruns and J. Gubeladze, Polytopes, rings and $K$-theory, Springer Mono. Math., Springer, Dorcrecht, 2009.
  • L. Reid, L.G. Roberts and B. Singh, Finiteness of subintegrality, NATO Adv. Sci. Inst. 407, Kluwer Academic Publishers, Dordrecht, 1993.
  • L.G. Roberts and B. Singh, Subintegrality, invertible modules and the Picard group, Compos. Math. 85 (1993), 249–279.
  • V. Sadhu, Subintegrality, invertible modules and Laurent polynomial extensions, arxiv 1404.6498, 2014.
  • V. Sadhu and B. Singh, Subintegrality, invertible modules and polynomial extensions, J. Algebra 393 (2013), 16–23.