Rings

Question 1

Unity and unit?

Answer 1

A unity is an identity element under multiplication.

A unit is an element that has a multiplicative inverse.

Question 2

Zero-divisor?

Answer 2

A nonzero element, a, of a commutative ring such that there exists a nonzero element b such that ab=0.

Question 3

Integral domain?

Answer 3

A commutative ring with unity and no zero-divisors.

Question 4

Field?

Answer 4

A commutative ring with unity such that every nonzero element is a unit.

Question 5

Characteristic of a ring?

Answer 5

The least positive integer n such that
nx=0 for all x in the ring.

If no such integer exists, then the characteristic is 0.

Question 6

Ideal?

Answer 6

A subring R’ of a ring R such that if R is in R and a is in R’, then ar and ra are in R’.

Question 7

Prime ideal?

Answer 7

A proper ideal A of a commutative ring R such that if a,b in R and ab in A, then either a or b is in A.

Question 8

Maximal ideal?

Answer 8

A proper ideal A of a commutative ring R such that if B is an ideal of R and
A subset B subset R, then A=B or B=R.

Question 9

What is a ring?

Answer 9

A collection with 2 binary operations such that:
It is an Abelian group under addition,
and
Multiplication is associative and left and right distributive over addition.