# rings and fields..urgent help plz

• Jan 29th 2007, 07:59 PM
jenjen
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.
• Jan 29th 2007, 08:03 PM
ThePerfectHacker
Quote:

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,
\$\displaystyle a^{-1}(ar)=a^{-1}(as)\$
Associate property,
\$\displaystyle (a^{-1}a)r=(a^{-1}a)s\$
\$\displaystyle 1r=1s\$
\$\displaystyle r=s\$

Quote:

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 :confused: ???

a+b=a+c
Then,
\$\displaystyle -a+(a+b)=-a+(a+c)\$
\$\displaystyle (-a+a)+b=(-a+a)+c\$
\$\displaystyle 0+b=0+c\$
\$\displaystyle b=c\$
• Jan 29th 2007, 08:23 PM
jenjen
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
• Jan 29th 2007, 08:28 PM
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.
• Jan 30th 2007, 09:33 AM
ThePerfectHacker
Quote:

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 :eek: