# Thread: rings and fields..urgent help plz

1. ## rings and fields..urgent help plz

Hi! I have a midterm tomorrow and I am stuck on this problem. Can anyone help me??? Thankkk youu soo much.

1) Suppose F is a field, and a is a nonzero element of F. Show that if r, s are in F and ar = as, then r = s.

2) Let R be a ring with identity. Prove that for all a, b, c in R, if a + b = 0 and a + c = 0 then b = c. (This means that for any a in R, the element -a is uniquely determined.

2. Originally Posted by jenjen
1) Suppose F is a field, and a is a nonzero element of F. Show that if r, s are in F and ar = as, then r = s.
Each non-zero element in a field has an inverse (by definition).
Thus,
$a^{-1}(ar)=a^{-1}(as)$
Associate property,
$(a^{-1}a)r=(a^{-1}a)s$
$1r=1s$
$r=s$

2) Let R be a ring with identity. Prove that for all a, b, c in R, if a + b = 0 and a + c = 0 then b = c. (This means that for any a in R, the element -a is uniquely determined.
What does it mean a ring with identity ???

a+b=a+c
Then,
$-a+(a+b)=-a+(a+c)$
$(-a+a)+b=(-a+a)+c$
$0+b=0+c$
$b=c$

3. Thank you for the quick reply ThePerfectHacker!!

Def: An element a of a ring with with identity R is called a unit of R if there exists some b in R so that a x b = b x a = 1

4. I also found this defintion

Def: A ring (with identity) is a set R with two operations, addition and mutiplication, and two special elements, 0 and 1, which satisfy axioms (associativity of addition, communtativity of addition, 0 is a zero element, etc.). The operations addition and multiplication may each be thought of as functions from R x R (ordered pairs of elements of the set R) to R, so that for any ordered pair (a,b), where a, b are in R, a + b is an element of R, and a x b is an element of R.

5. Originally Posted by jenjen
I also found this defintion

Def: A ring (with identity) is a set R with two operations, addition and mutiplication, and two special elements, 0 and 1, which satisfy axioms (associativity of addition, communtativity of addition, 0 is a zero element, etc.). The operations addition and multiplication may each be thought of as functions from R x R (ordered pairs of elements of the set R) to R, so that for any ordered pair (a,b), where a, b are in R, a + b is an element of R, and a x b is an element of R.
The standard terminology is a "ring with unity".
Perhaps you are using a text from the begining from the 20th Centrury