Rings, Integral Domains, Fields

Rings, integral domains, fields

Ring? (all rings Commutative? Integral domain? Field? are with unity) with + and * Yes Yes Yes No with + and * with + and * with + and *

M 2,2 with matrix + and *

Zp ( p prime) with + and * modulo p

Zm ( m composite) with + and * modulo m R[x ] (polynomials with Yes If R is If R is an integral Never coefficients in a ring) with commutative domain polynomial + and * F[x ] (polynomials with Yes Yes Yes No, never. coefficients in a field) with polynomial + and *

In addition to filling this out, this would also be a good place to list (perhaps on the back) 1. The definition and properties of a ring 2. What we mean by a commutative ring 3. What makes a ring an integral domain? 4. What makes an integral domain a field? 5. Related definitions (unity, zero divisor, unit) 6. The additive and multiplicative identities for each.