Ideals and The Spectrum
In the following, R denotes a commutative ring.In contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation. First, an element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. a non-zero element a such that there exists a non-zero element b of the ring such that ab = 0. If R possesses no zero divisors, it is called an integral domain since it closely resembles the integers in some ways.
Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably.
Read more about this topic: Commutative Ring
Famous quotes containing the word ideals:
“But I would emphasize again that social and economic solutions, as such, will not avail to satisfy the aspirations of the people unless they conform with the traditions of our race, deeply grooved in their sentiments through a century and a half of struggle for ideals of life that are rooted in religion and fed from purely spiritual springs.”
—Herbert Hoover (1874–1964)