Pool problems in ring theory

Consider the additive group of integers Z.

  1. Prove that every subgroup of Z is a cyclic group.
  2. Prove that every homomorphic image of Z is a cyclic group.
  3. Now consider the ring Z. Exhibit a prime ideal of Z that is not maximal.

Let R be an integral domain. Suppose that a and b are non-associate irreducible elements in R, and the ideal (a,b) generated by a and b is a proper ideal. Show that R is not a principal ideal domain (PID).


Let R be a commutative ring with identity. Suppose that for every aR there is an integer n2 such that an=a. Show that every prime ideal of R is maximal.


Let R be a commutative ring with 1, and σ:RR be a ring automorphism.

  1. Show that F={rRσ(r)=r} is a subring of R (with 1).
  2. Show that if σ2 is the identity map on R, then each element of R is the root of a monic polynomial of degree 2 in F[x], where F is as in (a).

A Boolean algebra is a ring A with 1 satisfying x2=x for all xA. Prove that in a Boolean algebra A:

  1. 2x=0 for all xA.
  2. Every prime ideal p is maximal, and A/p is a field with two elements.

Suppose R is a ring such that r2=r for every element rR.

  1. Prove r=r for every element rR.
  2. Show R must be commutative. Hint: Consider (a+b)2.

Let R be a commutative ring with 1. The characteristic char(R) of R is the unique integer n0 such that nZ is the kernel of the homomorphism θ:ZR given by

θ(m)={1R++1Rm, if m01R++1R|m|, if m<0

  1. Prove that if f:RS is a monomorphism of commutative rings with 1, then char(R)=char(S).
  2. Prove by given an example that char(R) is not always preserved by ring homomorphisms.

  1. Prove that for every commutative ring with unity, R, there is a unique ring homomorphism ϕR:ZR, and that ker(ϕR)=dR for some unique nonnegative integer dR. The number dR is called the characteristic of R and is denoted char(R).
  2. Suppose F1 and F2 are fields for which there exists a ring homomorphism f:F1F2. Prove that char(F1)=char(F2).

Let A be a commutative ring with 1. The dimension of A is the maximum length d of a chain of prime ideals p0p1pd. Prove that if A is a PID, the dimension of A is at most 1.


Prove that every Euclidean domain is a principal ideal domain.


Suppose R is a finite ring with no nontrivial zero-divisors. Prove that R contains an element 1 satisfying 1a=a1=a for all aR.


Let F be a field and let α be an element that generates a field extension of F of degree five. Prove that α2 generates the same extension.


Let R be a commutative ring with 1. Use theorems in ring theory to prove:

  1. x is a prime ideal in R[x] if and only if R is an integral domain.
  2. x is a maximal ideal in R[x] if and only if R is a field.

Let F be a field and F[x] be the polynomial ring, which is a principal ideal domain. Let R={fF[x]:f(x)}, where (x)F[x] is the ideal generated by x, and f is the (formal) derivative of the polynomial f. It is a fact that R is a subring of F[x].

  1. Prove that x2 and x3 are irreducible elements of R.
  2. Let (x2,x3) be the ideal in R generated by x2 and x3. Prove this is a proper ideal of R.
  3. Prove that (x2,x3) is not a principal ideal of R.

Let R be a commutative ring with 1 and suppose eR is idempotent, i.e., satisfies e2=e.

  1. Prove that 1e is also idempotent.
  2. Suppose e0,1. Show that Re and R(1e) are proper ideals of R.
  3. Prove there is an isomorphism RRe×R(1e).

An element r of a ring R is said to be idempotent if r2=r. Suppose that R is a commutative ring with unity containing an idempotent element e.

  1. Prove that 1e is also idempotent.

  2. Prove that eR and (1e)R are both ideals in R and that

    ReR×(1e)R.

  3. Prove that if R has a unique maximal ideal, then the only idempotent elements in R are 0 and 1.


Prove that if ϕ:RS is a surjective ring homomorphism between commutative rings with unity, then ϕ(1R)=1S.


Suppose R is a PID (principal ideal domain). Prove that an ideal IR is maximal if and only if I=p for a prime pR. (By definition, an element p is prime if whenever pab then pa or pb. If you use the fact that prime implies irreducible, you have to prove it.)


Let R be a commutative ring.

  1. Prove that the set N of all nilpotent elements of R is an ideal.
  2. Prove that R/N is a ring with no nonzero nilpotent elements.
  3. Show that N is contained in every prime ideal of R.

Let R be a commutative ring with 1. We say an element nR is nilpotent if there exists a number kN such that nk=0.

  1. Show that if n is nilpotent, then 1n is a unit.
  2. Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.

Let D be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.


Let IZ[x] denote the set of all polynomials with even constant term.

  1. Prove that I is an ideal.
  2. Prove that I is not a principal ideal.

Let R be a commutative ring with unity.

  1. Define what it means for an element in R to be prime, and also what it means for an element to be irreducible.
  2. Prove that if R is an integral domain, then every prime element is irreducible.

Let R be a commutative ring with unity, let IR be an ideal, and let π:RR/I be the natural projection homomorphism.

  1. Show that if is a prime ideal of R/I, then π1() is a prime ideal of R.
  2. Show that the assignment π1() is injective on the set of prime ideals of R/I.

Let A be a commutative ring with unit. We call A Boolean if a2=a for every aA. Prove that in a Boolean ring A each of the following holds:

  1. 2a=0 for every aA.
  2. If I is a prime ideal then A/I is a field with two elements (and in particular I is maximal).
  3. If I=(a,b) is the ideal generated by a and b then I can be generated by the single element a+b+ab. Conclude that every finitely generated ideal is principal.

Let R be a commutative ring. For each nonempty subset XR, the annihilator of X is the set ann(X)={aRax=0 for all xX}.

  1. Prove that ann(X) is an ideal of R.
  2. Prove that Xann(ann(X)).

Suppose ϕ:RS is a ring homomorphism, and S has no (nonzero) zero-divisors. Prove from the definitions that ker(ϕ) is a prime ideal.


Let R be a commutative ring with 1, and N the ideal

N={aRan=0 for some n}.

Let [b] be the image of bR in R/N. Prove that if [a]R/N and [a]m=0 then [a]=[0].


Let R1,,Rk be commutative rings, and set R=R1××Rk.

  1. Let IjRj be ideals, and put I=I1××Ik. Use the First Isomorphism Theorem to prove that R/IR1/I1××Rk/Ik.
  2. Prove the prime ideals of R have the form R1××Rj1×Pj×Rj+1××Rk where PjRj is a prime ideal for 1jk. (Omit the proof that this is an ideal.)

Let R be a commutative ring. The nilradical of R is defined to be N={rR|rn=0 for some nN}.

  1. Prove that N is an ideal of R.
  2. Prove that N is contained in the intersection of all prime ideals of R.

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