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Ring Abstract Algebra?
In each of the following cases, decide if the set with the given operations form a ring. If it is a ring prove all the properties of the ring and decide whether it is a commutative, has unity, has zero divisors, and is a field. If it is not a ring, show which property fails.
A) (3Z = {3k | k in Z}, +, x)
B) (Z x 2Z, +, x)
C) M2 (Z2), +, x) Note: M2 is a 2 by 2 matrix, Z2 is Z mod 2
Please explain in details. Because of hurricane Sandy my professor rush through this last chapter right before the final so our entire class is confused we form a study group, but no one knows what's going on with these questions. We need to understand these before Tuesday (December 18) Please help. Thank you for your help I really appreciated.
This is one of the problem from my text book and the answers are given, but I need to see the steps so I know how to get the answers. In the question where it says PROVE meaning use the Ring axioms to show that it is a ring or if it's not a ring then show which ring axiom fails.
THANK YOU!
1 Answer
- DavidLv 78 years agoFavorite Answer
A) is a ring, commutative, no unity, no zero divisors, is not a field.
B) is a ring, commutative, has unity (1,1), has zero divisors (a,0) and (0,1), is not a field (any field must be an integral domain, so having zero divisors rules that out).
C) is a ring, not commutative, has unity (the matrix I) has zero divisors (any matrix with 3 or more zeros, or a zero row or a zero column), is not a field.
i am not going to prove these satisfy the ring axioms, you should do this yourself.
