Maximal Ideals : Commutative Rings and Pointwise Operations ( Function Addition and Composition )

Let R be the set of all continuous functions from the set of real numbers into itself. Then R is a commutative ring with the following operations: (f+g)(x)=f(x) + g(x) and (fg)(x)=f(x)g(x) for all x.

Now let I be the set of all functions f(x) an element of R such that f(1)=0. Show that I is a maximal ideal of R.

Maximal ideal: I is a maximal ideal if for all ideals J of R such that I is contained in J is contained in R, then either J=I or J=R. In other words we cannot "squeeze" another ideal between I and the whole ring R.

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Proof:
First, I claim that I is an ideal.
For any f(x), g(x) in I, f(1)+g(1)=0+0=0 and f(1)g(1)=0*0=0. Thus I is a subring.
For any f(x) in I and g(x) in R, we have g(1)f(1)=0*g(1)=0. ...

Solution Summary

Maximal Ideals, Commutative Rings and Pointwise Operations ( Function Addition and Composition ) are investigated. The response received a rating of "5/5" from the student who originally posted the question.

Let phi:R->S be a homomorphism of commutativerings
a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R. Apply this to the special case when R is a subring of S and phi is the inclusion homomorphism to deduce that if P is a prime ideal of S then PR is either R or prime ideal in

If n Є R and R is a commutative ring we indicate by Mn(R) the ring of allnxn entries wrt the usual operations on matrices. If n>1 this ring is commutative even if R is.
Let S={(aij)ЄMn(R)|i≠j=>aij=0}
Let k be an integer 1≤k≤n. Show that
a) S is a commutative subring of Mn(R)
b) The function f: S

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 maximalideals 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

(1) Given a ring R, an element e is called an idempotent if e^2 = e.
(i) Let R1 and R2 be two commutativerings with unity. Consider R = R1 x R2. Find two non-zero idempotents e1,e2 E R such that 1= e1 + e2 and e1e2 = 0. (Be careful: what is 1 in this R? What is 0?)
(ii) On the other hand, suppose R is any commutative ring wit

Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0).
If f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor, prove that there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0.
See attached file for full problem description.

If R is an integral domain, then so is R[x].
Prove that if R is an integral domain, then R[x] is also an integral domain.
See attached file for full problem description.

(See attached file for full problem description with proper symbols)
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1A) Let R be a commutative ring and let A = {t R tp = 0R} where p is a fixed element of R. Prove that if k, m A and b R, then both k + m and kb are in A.
1B) Let R be a commutative ring and let b be a fixed ele