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A group with an extra operation called multiplication which satisfies:
Unlike addition, that multiplication does not need to satisfy:
If those are also satisfied, then we have a field.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set.
A polynomial ring is another example with the same properties as the integers.
The simplest non-commutative ring that is not a field is the set of all 2x2 matrices of real numbers:
  • we know that 2x2 matrix multiplication is non-commutative in general
  • some 2x2 matrices have a multiplicative inverse, but others don't
Note that is not a ring because you can by addition reach the zero matrix.