Arithmetic is the art of manipulating terms by the basic operations of addition, subtraction and multiplication and also division whenever possible. This art can be practised in different domains; we can not only occupy ourselves with integers or rational numbers, but also with polynomials, matrices or other objects. In all these domains, the used operations satisfy certain arithmetic laws which we usually use without thinking. Now we will axiomatize the notions of addition and multiplication; namely, we will define a ring as a set with two arithmetic operations, called addition and multiplication, which satisfy certain rules (the “arithmetic laws”). This definition will turn out to be “weak” enough to include a variety of diverse examples, but “strong” enough to allow interesting conclusions and non-trivial applications. As guiding examples, we should keep in mind the set ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/0842885d-d52d-49f7-87c1-c87f0884069c/content/eq97.tif"/> of all integers, the set ℤ [ x ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/0842885d-d52d-49f7-87c1-c87f0884069c/content/eq98.tif"/> of all polynomials with integer coefficients and the set ℤ n × n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/0842885d-d52d-49f7-87c1-c87f0884069c/content/eq99.tif"/> of all n × n-matrices with integer entries; each endowed with the usual addition and multiplication.