1. Let R be a ring and let I be an ideal in R. Prove using the definition of coset addition and properties of R that the quotient ring satisfies the associative axiom for addition.
Let . Then
2. (a) Find the number of elements in the quotient ring .
(b) Find all the roots of in . Prove or disprove: is a field.
(a) Every coset can be simplified to the form , where . Hence, the quotient ring has elements.
has roots, so it's not irreducible, and is not a field.
3. Consider the quotient ring .
(a) How many elements are there in ?
(b) Compute . Simplify your answer to the form , where .
(c) Compute . Simplify your answer to the form , where .
(d) Compute . Simplify your answer to the form , where .
(a) Since every element can be written as , where , there are elements.
4. A ring R (not necessarily commutative or having an identity element) is Boolean if for all .
(a) Prove that in a Boolean ring, for all .
(b) Prove that a Boolean ring is commutative.
(a) Let R be a Boolean ring and let . By assumption, . Expanding the left side, I have
(b) Let R be a Boolean ring and let . By assumption, . Expanding the left side, I have
The last step used the result of part (a) (which shows that ).
Mingle some brief folly with your wisdom. - Horace
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