Question

What is the difference between a domain, a group, a field, and a ring?

Answer 1

also, a space.

Answer 2

A field (A,+,*) is an algebraic structure, which is closed upon + and *, where + and * are associative, commutative, and + and * have each a neutral element, (these two element must be different), where a additive and a multiplicatove identity holds, and finally where * distrivbutes over +.

Answer 3

A ring (A,+,*) is such that (A,+) is an abelian group (neutral element 0), and (A,*) is a monoid (neutral element 0, 0!= 1).

Answer 4

An abelian group (A,+) is a stuctrure such that A is closed upon +; + is associative, commutative, has a neutral element, has a inverse element.

Answer 5

A non-abelian group (simply group) is a abelian group, but + does not need to be commutative. A monoid is a group without inverse element. A semigroup is a monoid without neutral element. A magma is a semigroup without associativity. A set is a degeranted magma, without composition law.

Answer 6

A (integral) domain is more or less a set defined as an interval.. Last bu not least, a field needs to be not a trivial ring (the unary ring is not a field.) That's all.