2) Special Types of Rings and Elements

Question 1

What is the characteristic of a ring

Answer 1

The least positive integer n such that
n · 1 = 1 + · · · + 1| {z }n= 0
If there is no such n, then the characteristic of R is defined to be 0

Question 2

What are the implications of a ring with char(R)=n>0

Answer 2

(i) n · r = 0 for every r ∈ R
(ii) if m is a positive integer then m · 1 = 0 iff n|m

Question 3

Describe the proof of the characteristic properties of a ring with char(R)=n>0

Answer 3

(i) n · r = r + … + r= r(1 + … + 1) =
r × 0 = 0
(ii)Let m ∈ N with m · 1 = 0 and write m = nq + r where 0 ≤ r ≤ n − 1 and q ≥ 0 are integers. Then
0 = (nq + r) · 1 = q · (n · 1) + r · 1 = 0 + r · 1 = r · 1
By the minimality of n, we must have r = 0 and so n|m.
Conversely if n|m, then m = nq and so m · 1 = (nq) · 1 = q · (n · 1) = 0

Question 4

What is a zero-divisor

Answer 4

A non-zero element r is a zero-divisor if there is a non-zero element s ∈ R with rs = 0 or sr = 0

Question 5

What is a domain

Answer 5

A ring R is a domain if, for all r, s ∈ R
rs = 0 ⇒ r = 0 or s = 0
Therefore a domain is a ring with no zero-divisors

Question 6

What is an integral domain

Answer 6

A commutative domain

Question 7

If S is a subring of R and R is a domain, what can be said about S

Answer 7

S is a domain

Question 8

Describe the proof that if S is a subring of R and R is domain then S is a domain

Answer 8

Let r, s ∈ S with rs = 0. Then rs = 0 in R and so r = 0 or s = 0

Question 9

What is the implication of R being a domain on its polynomial ring R[X]

Answer 9

Then the polynomial ring R[X] is a domain

Question 10

Is the Cartesian Product of two domains, a domain

Answer 10

No, (1, 0)(0, 1) = (0, 0) and so R × S has zero divisors

Question 11

Describe the proof that if R is a domain. Then the polynomial ring R[X] is a domain

Answer 11

Question 12

What is the implication for the ring R[X1,…,Xn ] when R is a domain

Answer 12

The ring, R[X1, . . . , Xn], of polynomials in n indeterminates and with coefficients in R, is a domain

Question 13

Prove that if R is a domain then char(R) = 0 or char(R) is a prime
integer.

Answer 13

If char(R) = 0 we are done, so assume that char(R) = n > 0.
If n is not prime we can factorise: n = kl with 1 < k, l < n. Then 0 = n · 1 = (k · 1)(l · 1)
Since R is a domain this means that either k · 1 = 0 or l · 1 = 0, contradicting the minimality of n

Question 14

What is a division ring

Answer 14

A ring in which every non-zero element has a right inverse and a left inverse

Question 15

What is a unit

Answer 15

An any non-zero element in a division ring that is invertible, having a unique inverse within the ring

Question 16

What is a field

Answer 16

A commutative division ring

Question 17

What happens if an element r in a ring R has both a right and a left inverse

Answer 17

Then these are equal and unique

Question 18

Describe the proof that if a ring has both a right and a left inverse then these are equal and unique

Answer 18

Let s be the right inverse and t be the left inverse of r. Consider trs:
trs = (tr)s = 1s = s
trs = t(rs) = t1 = t.
So s = t

Question 19

What caution must be observed regarding inverses in non-commutative rings

Answer 19

• In non-commutative rings, an element can have a left inverse without having a right inverse
• When an element has a left inverse but no right inverse, it might possess more than one left inverse

Question 20

What conditions are equivalent for an integer n≥2

Answer 20

Question 21

What is the relationship between division rings, fields, and domains

Answer 21

• Every division ring is a domain
• Every field is an integral domain

Question 22

Describe the proof that every division ring is a domain

Answer 22

Let R be a division ring and suppose r, s ∈ R with rs = 0
Assume r ≠ 0. Then r^−1 exists and 0 = r^−1(rs) = (r
−1r)s = 1s = s
Therefore R contains no divisors of zero and so is a domain

Question 23

What is the nature of the set of units R∗ in a ring R

Answer 23

In any ring R, the set of units R∗ forms a a group with multiplication

Question 24

What does it mean for an element to be nilpotent

Answer 24

An element r of a ring R is nilpotent if there is some integer n ≥ 1 with r^n = 0

Question 25

What is the the index of nilpotence

Answer 25

The least such n that r^n = 0

Question 26

What does it mean for an element to be idempotent

Answer 26

r^2 = r