High Quality Content by WIKIPEDIA articles! In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication. It is a commutative ring, and is the prototypical such by virtue of satisfying only those equations holding of all commutative rings with identity; indeed it is the initial commutative ring, as well as being the initial ring.More generally the ring of integers of an algebraic number field K, often denoted by OK (or mathcal O_K), is the ring of algebraic integers contained in K.Using this notation, we can write Z = OQ since Z as above is the ring of integers of the field Q of rational numbers. And indeed, in algebraic number theory the elements of Z are often called the "rational integers" because of this. An alternative term is maximal order, since the ring of integers of a number field is indeed the unique maximal order in the field.

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