ARTINIAN SUBRINGS OF PRODUCT OF ZERO-DIMENSIONAL RINGS

ABSTRACT

Given a family of commutative rings {R _α}_α∈A , we investigate the family 𝒜(∏_α∈A R _α) of Artinian subrings of ∏_α∈A R _α. This paper deals also with the question of when a pair (∏_α∈A R _α, ∏_α∈A T _α) is zero-dimensional, where {(R _α, T _α)}_α∈A is a family of zero-dimensional pairs.

Key Words:

• Artinian ring
• Artinian subring
• Hereditarily zero-dimensional
• Infinite product of rings
• Quasi-local ring
• Zero-dimensional pair
• Zero-dimensional ring

2000 Mathematics Subject Classification:

• Primary 13A99
• Secondary 13A15
• 13B02
• 13E05

ACKNOWLEDGMENT

I would like to thank Prof. L. Izelgue for his valuable comments during the preparation of this paper. As well, I gratefully acknowledge the helpful comments made by the referee of this paper.

Notes

¹ R is said to be hereditarily zero-dimensional if each subring of R is zero-dimensional.

^#Communicated by A. Facchini.