Chapter 2 Special Types of Rings and Elements

Question 1

What is the characteristic of a ring?

Answer 1

The CHARACTERISTIC Char(R) of a ring is the least positive integer n such that n•I=0 (I subscript R)
If no such n exists we say Char(R)=0

Question 2

Define n•a

Answer 2

n•a= a+…+a (n times)

We can extend this definition by using 0•a=0 and n•a=(-n)•(-a)

Question 3

What is a Zero Divisor?

Answer 3

A non zero element, r in R, is a ZERO divisor if there is another nonzero element s in R with either sr=0 or rs=0

Question 4

What does it mean for a ring to be a domain?

Answer 4

A ring is a DOMAIN if it has no zero divisors for all r and s in R

Question 5

What is a division ring?

Answer 5

A DIVISION RING is a ring in which every non-zero element has a right inverse and a left inverse wrt X

Question 6

TFAE: Z is an integral domain

Answer 6

n is prime

Z_n is a field

Question 7

What conditions does an element satisfy to be nilpotent?

Answer 7

An element r in a ring R is NILPOTENT if there exists a positive integer n such that r^n=0

Question 8

What is the nilpotence of R?

Answer 8

the lowest n such that An element r in a ring R is Nilpotent if there exists a positive integer n such that r^n=0

Question 9

What conditions does an element satisfy to be idempotent?

Answer 9

An element is IDEMPOTENT if r^2=r