Basic Ring Theory

Question 1

Group

Answer 1

A set G with a binary operation • : G x G → G with

• associativity,
• unique inverses,
• unique identity.

Question 2

Ring

Answer 2

A set R with two binary operations +, • st

• (R,+) is an abelian group,
• (R,•) is a semigroup, ie.
  
  • • is associative, and
  • there exists 1 in R st 1 • a = a • 1 = a
• (R,+,•) satisfies the distributive laws.4

All rings here are commutative.

Question 3

Ideal

Answer 3

A nonempty subset I of R is an ideal if

-a, a+b, ra are in I, for all a,b in I, r in R.

An ideal I is principle if I = Ra = { ra | r in R }, ( I is generated by a).

Question 4

Integral domain

Answer 4

A commutative nontrivial ring R is an integral domain if

rs = 0 ⇒ r = 0 or s = 0.

Question 5

Principle ideal domain

Answer 5

R is commutative, an ID and any ideal of R is principle.

Question 6

Field

Answer 6

R is commutative and ( R{0}, • ) is a group.

Question 7

Subring

Answer 7

A subset S of R is a subring of R if

• ( S, + ) is a subgroup, and
• S is closed under multiplication and contains the identity.

Question 8

Subfield

Answer 8

A subring S of a field F is a subfield of F if S is closed under taking multiplicative inverses.