Math 361, Spring 2015, Assignment 4
From cartan.math.umb.edu
Carefully define the following terms, then give one example and one non-example of each:
- Kernel (of a ring homomorphism).
- Ideal.
- Principal idea.
- Quotient ring (a.k.a. factor ring).
- Prime ideal.
- Maximal ideal.
Carefully state the following theorems (you do not need to prove them):
- Theorem concerning images and pre-images of subrings.
- Theorem characterizing the subrings that can be kernels of homomorphisms.
- Fundamental theorem of ring homomorphisms.
- Equivalent conditions for containment of principal ideals.
- Equivalent condition for equality of principal ideals.
- Theorem characterizing prime ideals (in terms of properties of the corresponding quotient rings).
- Theorem characterizing maximal ideals (in terms of properties of the corresponding quotient rings).
- Theorem relating maximal ideals to prime ideals.
Solve the following problems:
- Section 26, problems 3, 11, 12, 13, and 14.
- Section 27, problems 1, 10, 15, 16, and 17.
Questions:
Does anyone know the theorem number corresponding to "Theorem characterizing the subrings that can be kernels of homomorphisms" in the book?
Solutions:
Definitions:
- The Kernel of a ring morphism ϕ is the subring S such that ϕ[S]={0}.
- An Ideal I of a ring R is a subring of R such that for all i∈I,a∈R, ai∈R and ia∈R, that is to say I preserves right and left products.
- If R is a commutative unital ring and a∈R, the ideal {ra|r∈R} of all multiples of a is the Principal Ideal Generated By a and is denoted by ⟨a⟩. An ideal I of R is a Principal Ideal if I=⟨a⟩ for some a∈R.
- Let I be an ideal of a ring R. Then the additive cosets of I form the Quotient Ring R/I under the binary operations defined by:
- (a+I)+(b+I)=(a+b)+I
- (a+I)(b+I)=ab+I
- An ideal I≠R in a commutative ring R is a Prime Ideal if ab∈I implies a∈I or b∈I for a,b∈R.
- A Maximal Ideal of a ring R is an ideal M≠R such that there is no proper ideal I of R with I properly conaining M.
Theorems:
- Let S,R be rings and let ϕ:R→S be a ring morphism. Let A be a subring of R and let B be a subring of S. Then:
- ϕ[A] is a subring of S and
- ϕ−1[B] is a subring of R
- Let ϕ:R→R′ be a ring morphism with kernel N. Then ϕ[R] is a ring, and the map μ:R/N→ϕ[R] given by μ(x+N)=ϕ(x) is an isomorphism. If γ:R→R/N is the morphism given by γ(x)=x+N, then for each x∈R we have ϕ(x)=μγ(x).
- Let D be a domain. The following statements are equivalent for all a,b∈D:
- ⟨a⟩⊆⟨b⟩
- a∈⟨b⟩
- b|a
- ⟨a⟩=⟨b⟩ if and only if a∼b.
- An ideal I of a ring R is prime if and only if R/I is a domain.
- An ideal I of a ring R is maximal if and only if R/I is a field.
- Evey maximal ideal is a prime ideal.