Various Facets of Rings Between D[X] and K[X]

Abstract

Since the circulation, in 1974, of the first draft of “The construction D + XD _S [X], J. Algebra 53 (1978), 423–439” a number of variations of this construction have appeared. Some of these are: The generalized D + M construction, the A + (X)B[X] construction, with X a single variable or a set of variables, and the D + I construction (with I not necessarily prime). These constructions have proved their worth not only in providing numerous examples and counter examples in commutative ring theory, but also in providing statements that often turn out to be forerunners of results on general pullbacks. The aim of this paper will be to discuss these constructions and the remarkable uses they have been put to. I will concentrate more on the A + XB[X] construction, its basic properties and examples arising from it.

Key Words:

• Polynomial ring
• A + XB[X]
• D + M
• Pullback
• Composite

1991 Mathematics Subject Classification:

• Primary 13B25
• 13F20
• Secondary 13A05
• 13A18

Acknowledgments

David Anderson has so kindly read an earlier version of this article. His advice has always been useful to me. Dan Anderson and Joe Mott also offered advice that I am grateful for. Tiberiu Dumitrescu and Lahoucine Izelgue have also read parts of the script. I am thankful to all. It is indeed fair to say that it would be impossible for me to have done as much as I was able to do without help and support from my friends and benefactors. If, in spite of all that help, there are any mistakes, they are all mine. Finally, I thank the referee for pointing out several formatting errors and several typographical errors.

Notes

^#This article was to have appeared in full in the recent Dekker publication, “Commutative Ring Theory and Applications,” in Volume 231 of the Dekker Lecture Notes series; however, part of the article was inadvertently omitted from that volume. The editors of the Dekker volume and Communications in Algebra agreed to remedy the problem by republishing this article, which was originally received in October 2001.