Monoids, Groups, Rings and Fields

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 5 June 2012

Monoids, Groups, Rings and Fields

Examples.

  1. (a) The positive integers >0 with the addition operation is a monoid without identity.
  2. (b) The nonnegative integers 0 with the addition operation is a monoid.
  3. (c) The integers with the addition operation is an abelian group.
  4. (d) The integers with the addition and multiplication operations is a commutative ring.
  5. (e) The rationals with the operations addition and multiplication is a field.
  6. (f) The quaternions with the oeprations of addition and multiplication form a division ring that is not a field.
  7. (g) The 2×2 matrices with entries in , M2(), with the operations of addition and multiplication of matrices is a ring which is not commutative.
  8. (h) The set of invertible 2×2 matrices with entries in , GL2(), with the operation of multiplication of matrices is a group which is not an abelian group.

Notes and References

Welcome to our (algebraic) zoo. This page tells you the types of animals in our zoo.

The unusual notations ?, !, and y# used in the above definitions of monoid and group are there to make the point that the usual notations (of i+j for addition of i and j, ij for multiplication of i and j, of 0 for the additive identity, of 1 for the mutliplicative identity, of -y for the additive inverse, and of y-1 for the multiplicative inverse of y) are rather arbitrary. But habits are important and useful and often efficient and helpful for clear communication, and so it is better to use the standard notations unless there is a very good reason not to.

These definitions are found in [Bou, Alg. Ch. I].

References

[Bou] N. Bourbaki, Algebra, Masson????? MR?????.

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