Chapter 7 - Introduction to Rings Flashcards
Define a ring
A ring R is a set together with two binary operations + and * where:
(i) (R,+) is an Abelian group
(ii) * is associative
(iii) The distributive laws hold in R: for all a, b, c in R, we have that (a+b)c = ac+bc and a(b+c) = ab+ac
What does it mean to say a ring is commutative?
Multiplication is commutative in the ring
What is an Abelian group?
An Abelian group is a group where addition is commutative.
What does it mean to say a ring has identity 1?
If there is some element 1 in R such that for all a in R, 1a = a1 = a
What is a division ring (aka skew field)?
A ring with identity 1 neq 0 is where any nonzero element a in R has a multiplicative inverse b in R such that ab = ba = 1.
What is a field?
A commutative division ring
We have four useful properties about any ring R:
(1) 0a = _ = _ for all a in R
(2) (-a)b = _ = _ for all a and b in R
(3) (-a)(-b) = _ for all a and b in R
(4) If R has a 1, then is that 1 unique? Also -a = (_)(a)?
(1) 0a = a0 = 0
(2) (-a)b = b(-a) = -(ab)
(3) (-a)(-b) = ab
(4) If R has a 1, then that 1 is unique and -a = (-1)(a).
What is a zero divisor of a ring?
A zero divisor is a nonzero element a in R such that there is some b in R for which ab = 0 or ba = 0.
What is a unit of a ring with identity 1 neq 0?
A unit is an element a in R such that there is some b in R for which ab = ba = 1.
What is R^x?
R^x is the set of units of R.
Can a zero divisor ever be a unit?
No, a zero divisor can never be a unit.
Do fields contain zero divisors? Units?
Fields contain no zero divisors, and in a field, every nonzero element is a unit.
What is an integral domain?
An integral domain is a commutative ring R with identity 1 neq 0 such that R contains no zero divisors.
Integral domains have a cancellation property. What does this mean?
For elements a, b, and c in R with a not a zero divisor:
ab = ac => a = 0 or b = c.
Are finite integral domains fields?
Yes, every finite integral domain is a field
