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

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• 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

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• 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