Deductive varieties of modules and universal algebras
HTML articles powered by AMS MathViewer
- by Leslie Hogben and Clifford Bergman PDF
- Trans. Amer. Math. Soc. 289 (1985), 303-320 Request permission
Abstract:
A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above.References
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
- Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR 648287
- David M. Clark and Peter H. Krauss, Varieties generated by para primal algebras, Algebra Universalis 7 (1977), no. 1, 93–114. MR 429696, DOI 10.1007/BF02485419
- W. Edwin Clark and Joseph J. Liang, Enumeration of finite commutative chain rings, J. Algebra 27 (1973), 445–453. MR 337910, DOI 10.1016/0021-8693(73)90055-0
- Nathan Divinsky, Rings and radicals, Mathematical Expositions, No. 14, University of Toronto Press, Toronto, Ont., 1965. MR 0197489 R. S. Freese and R. N. McKenzie, The commutator, an overview preprint, 1981.
- H.-Peter Gumm, An easy way to the commutator in modular varieties, Arch. Math. (Basel) 34 (1980), no. 3, 220–228. MR 590312, DOI 10.1007/BF01224955
- Christian Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 119–125. MR 534504
- I. N. Herstein, Topics in algebra, 2nd ed., Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. MR 0356988
- V. I. Igošin, Quasivarieties of lattices, Mat. Zametki 16 (1974), 49–56 (Russian). MR 360383
- Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, 190 Hope Street, Providence, R.I., 1956. MR 0081264
- Nathan Jacobson, Basic algebra. II, W. H. Freeman and Co., San Francisco, Calif., 1980. MR 571884
- J. Ježek and T. Kepka, Free commutative idempotent abelian groupoids and quasigroups, Acta Univ. Carolin. Math. Phys. 17 (1976), no. 2, 13–19 (English, with Russian and Czech summaries). MR 422479
- Ralph McKenzie, Narrowness implies uniformity, Algebra Universalis 15 (1982), no. 1, 67–85. MR 663953, DOI 10.1007/BF02483709
- K. R. McLean, Commutative artinian principal ideal rings, Proc. London Math. Soc. (3) 26 (1973), 249–272. MR 319981, DOI 10.1112/plms/s3-26.2.249
- Robert W. Quackenbush, Algebras with minimal spectrum, Algebra Universalis 10 (1980), no. 1, 117–129. MR 552161, DOI 10.1007/BF02482895
- Walter Taylor, Some applications of the term condition, Algebra Universalis 14 (1982), no. 1, 11–24. MR 634412, DOI 10.1007/BF02483903
- Kôshichi Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221–227. MR 14105
- A. I. Citkin, Structurally complete superintuitionistic logics, Dokl. Akad. Nauk SSSR 241 (1978), no. 1, 40–43 (Russian). MR 510889
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 289 (1985), 303-320
- MSC: Primary 08C15; Secondary 16A35
- DOI: https://doi.org/10.1090/S0002-9947-1985-0779065-X
- MathSciNet review: 779065