Condensed Rings with Zero-Divisors^#

^#To David Dobbs on his Sixtieth Birthday

ABSTRACT

A commutative ring R with identity is condensed (respectively strongly condensed) if for each pair of ideals I, J of R, IJ = {ij | i ∈ I, j ∈ J} (resp., IJ = iJ for some i ∈ I or IJ = Ij for some j ∈ J). In a similar fashion we can define regularly condensed and regularly strongly condensed rings by restricting I and J to be regular ideals. We show that an arbitrary product of rings is condensed if and only if each factor is so, and that R[X] is condensed if and only if R is von Neumann regular. A number of results known in the domain case are extended to the ring case. Regularly strongly condensed and one-dimensional regularly condensed Noetherian rings are characterized.

Key Words:

• Condensed ring
• Strongly condensed ring

2000 Mathematics Subject Classification:

• Primary 13A15
• Secondary 13F99

Notes

^#To David Dobbs on his Sixtieth Birthday

Communicated by J. Kuzmanovich.