Zero-dimensionality in commutative rings

Gilmer, Robert; Heinzer, William

doi:10.1090/S0002-9939-1992-1095223-0

Zero-dimensionality in commutative rings
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by Robert Gilmer and William Heinzer PDF

Proc. Amer. Math. Soc. 115 (1992), 881-893 Request permission

Abstract:

If ${\left \{ {{R_\alpha }} \right \}_{\alpha \in A}}$ is a family of zero-dimensional subrings of a commutative ring $T$, we show that ${ \cap _{\alpha \in A}}{R_\alpha }$ is also zero-dimensional. Thus, if $R$ is a subring of a zero-dimensional subring $[unk]\;T$ (a condition that is satisfied if and only if a power of $rT$ is idempotent for each $r \in R$, then there exists a unique minimal zero-dimensional subring ${R^0}$ of $T$ containing $R$. We investigate properties of ${R^0}$ as an $R$-algebra, and we show that ${R^0}$ is unique, up to $R$-isomorphism, only if $R$ itself is zero-dimensional.

References

M. Arapović, Characterizations of the $0$-dimensional rings, Glasnik Mat. Ser. III 18(38) (1983), no. 1, 39–46 (English, with Serbo-Croatian summary). MR 710383
M. Arapović, Characterizations of the $0$-dimensional rings, Glasnik Mat. Ser. III 18(38) (1983), no. 1, 39–46 (English, with Serbo-Croatian summary). MR 710383
Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR 0427289
Robert Gilmer and William Heinzer, On the imbedding of a direct product into a zero-dimensional commutative ring, Proc. Amer. Math. Soc. 106 (1989), no. 3, 631–636. MR 969521, DOI 10.1090/S0002-9939-1989-0969521-2
Robert Gilmer and William Heinzer, Products of commutative rings and zero-dimensionality, Trans. Amer. Math. Soc. 331 (1992), no. 2, 663–680. MR 1041047, DOI 10.1090/S0002-9947-1992-1041047-4
Robert Gilmer and William Heinzer, Artinian subrings of a commutative ring, Trans. Amer. Math. Soc. 336 (1993), no. 1, 295–310. MR 1102887, DOI 10.1090/S0002-9947-1993-1102887-7
Sarah Glaz, Commutative coherent rings, Lecture Notes in Mathematics, vol. 1371, Springer-Verlag, Berlin, 1989. MR 999133, DOI 10.1007/BFb0084570
James A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, vol. 117, Marcel Dekker, Inc., New York, 1988. MR 938741
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Paolo Maroscia, Sur les anneaux de dimension zéro, Atti Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 56 (1974), no. 4, 451–459 (French, with Italian summary). MR 0389877

Anneaux absolument plats universels et epimorphismes a buts reduits

Nicolae Popescu and Constantin Vraciu, Some remarks about the regular ring associated to a commutative ring, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 2, 269–277. MR 472899
Roger Wiegand, Descent of projectivity for locally free modules, Proc. Amer. Math. Soc. 41 (1973), 342–348. MR 327737, DOI 10.1090/S0002-9939-1973-0327737-0
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876