A Note on rings of finite rank

ABSTRACT

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matsuda and Matson showed that every n∈ℤ⁺ (the positive integers) occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n∈ℤ⁺). Two constructions use one-dimensional domains. Our third construction uses Artinian rings (dimension zero), and our last construction uses polynomial rings over local Artinian rings (dimension one, irreducible, not a domain).

KEYWORDS:

• Finite rank
• nonmaximal order
• semigroup ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13E15

Acknowledgements

Thanks to K. Conrad, P. Pollack and L.D. Watson for useful conversations.

Notes

¹Here all rings are commutative and with multiplicative identity.

²We are permitted to take α₁ = 1 by “Hermite’s Lemma” [2, Prop. 6.14].

³The hypothesis that 𝔯 not be a field does not explicitly appear in Matsuda’s work, but it must be intended.