Thread: 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.

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,

Associate property,

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,

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

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.

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