A ring R is called a Boolean ring if a^2=a for all a?R. Let R=P(X)be the power set of X. Define addition and multiplication in R as follows:
a+b=(a?b^')?(a^'?b)
a×b=a?b
Show that (R,+,*) is a Boolean ring.

Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1).
Or, Prove that the element 'a' in the Euclidean ring is a unit if and only if d(a) = d(1).

Prove that in a Euclidean ring ( a , b ) can be found as follows
b = q0 a + r1 where d ( r1) < d (a )
a = q1 r1 + r2 where d ( r2) < d (r1 )
r1 = q1 r1 + r2 where d ( r3) < d ( r2 )

Let F be the set of all functions f : RR. We know that is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x). We define multiplication on F by (fg)(x) = f(x)g(x). That is, fg is the function whose value at x is f(x)g(x). Show that the multiplication defined on the set F satisfies axio

With respect to the ideal
I=<2,x> in Z[x]
I believe this ideal is maximal because one theorem I have read suggests to me that all maximal ideals of Z[x] are in the form

where p is prime and f(x) is an element of Z[x] and irreducible mod p. It appears that <2,x> fits this description.
Did I understand correc

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Let R1 and R2 be integral domains with quotient fields F1 and F2 respectively. If phi: R1 -> R2 is a ring isomorphism, show that phi extends to an isomorphism phi hat : F1 -> F2. Here extends means that phi hat(a) = phi (a) for all a in R1 (Hint: under the givien assumptions, there is really only one way to

Prove that the equivalence relation modulo m where m is an integer, forms a ring.
Also, does this same equivalence relation form a field and why?
For this proof, you are given that [a]m (m is a subscript) represents an equivalence class modulo m, where m is an integer. We also know that for any two integers, a and b, that

I am stuck with this question.
If phi : F -> R is a nonzero homomorphism from a field F to a ring R, show that phi is one-to-one (Hint: recall that for a ring homomorphism, phi is one-to-one if and only if Ker(phi)={0}. )
Can you help with this?
Many thanks in advance