Rengasteoria

Rengasteoria on algebran osa-alue, jonka tutkimuskohteena ovat renkaat – algebralliset rakenteet, joille on määritelty yhteen- ja kertolasku.

Vaihdannaiset renkaat

Pääartikkelit: Vaihdannainen algebra ja vaihdannainen rengas

Rengasta kutsutaan vaihdannaiseksi, jos sen kertolasku on vaihdannainen. Jos ${\textstyle R}$ on rengas, jossa on määritelty yhteenlasku (${\textstyle +}$) ja kertolasku (${\textstyle \cdot }$), niin ${\textstyle R}$ on vaihdannainen, jos mille tahansa alkioille ${\textstyle a,b\in R}$ pätee $a\cdot b=b\cdot a$. Esimerkkejä vaihdannaisista renkaista ovat polynomirenkaat, algebrallisten kokonaislukujen renkaat samoin kuin tavalliset kokonaisluvut, ja p-aditiset kokonaisluvut.

Kirjallisuutta

Suomeksi

• Häsä, Jokke; Rämö, Johanna: Johdatus abstraktiin algebraan. Helsinki: Gaudeamus, 2015. ISBN 978-952-495-361-0.

Englanniksi

• Allenby, R. B. J. T. (1991), Rings, Fields and Groups (Second ed.), Edward Arnold, London, p. xxvi+383, ISBN 0-7131-3476-3, MR 1144518

• Blyth, T.S.; Robertson, E.F. (1985), Groups, Rings and Fields: Algebra through practice, Book 3, Cambridge: Cambridge University Press, ISBN 0-521-27288-2
• Faith, Carl (1999), Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, 65, Providence, RI: American Mathematical Society, ISBN 0-8218-0993-8, MR 1657671

• Goodearl, K. R.; Warfield, R. B., Jr. (1989), An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, 16, Cambridge: Cambridge University Press, ISBN 0-521-36086-2, MR 1020298

• Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications

• Kimberling, Clark (1981), "Emmy Noether and Her Influence", in Brewer, James W; Smith, Martha K, Emmy Noether: A Tribute to Her Life and Work, Marcel Dekker, pp. 3–61

• Lam, T. Y. (1999), Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 0-387-98428-3, MR 1653294

• Lam, T. Y. (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131 (Second ed.), New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439

• Lam, T. Y. (2003), Exercises in Classical Ring Theory, Problem Books in Mathematics (Second ed.), New York: Springer-Verlag, ISBN 0-387-00500-5, MR 2003255

• Matsumura, Hideyuki (1980), Commutative Algebra, Mathematics Lecture Note Series, 56 (Second ed.), Reading, Mass.: Benjamin Cummings, ISBN 0-8053-7026-9, MR 0575344

• McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30, Providence, RI: American Mathematical Society, doi:10.1090/gsm/030, ISBN 0-8218-2169-5, MR 1811901

• O'Connor, J. J.; Robertson, E. F. (September 2004), "The development of ring theory", MacTutor History of Mathematics Archive

• Pierce, Richard S. (1982), Associative Algebras, Graduate Texts in Mathematics, 88, New York: Springer-Verlag, ISBN 0-387-90693-2, MR 0674652

• Rowen, Louis H. (1988), Ring Theory, Vol. I, Pure and Applied Mathematics, 127, Boston, MA: Academic Press, ISBN 0-12-599841-4, MR 0940245. Vol. II, Pure and Applied Mathematics 128, ISBN 0-12-599842-2.

• Weibel, Charles A. (2013), The K-book: An introduction to algebraic K-theory, Graduate Studies in Mathematics, 145, Providence, RI: American Mathematical Society, ISBN 978-0-8218-9132-2, MR 3076731