Ideals, Ring Homomorphisms and Quotient Rings
Handbook guide to ideals, ring homomorphisms and quotient rings with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Ideals, Homomorphisms, and Quotient Rings
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
If f : R →S, where R and S are rings, call f is a ring homomorphism if f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b) for all a, b ∈R, and f(1R) = 1S.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Example
Let f : Z →Mn(R), n ≥2, be defined by f(n) = nE11 (see (2.1.3), example 3). Then one has f(a + b) = f(a) + f(b), f(ab) = f(a)f(b), but f(1) ̸= In. Thus f is not a ring homomorphism. In Chapter 1, we proved the basic isomorphism theorems for groups, and a key observation was the connection between group homomorphisms and normal subgroups.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Definition
Let I be a subset of the ring R, and consider the following three properties: (1) I is an additive subgroup of R (2) If a ∈I and r ∈R then ra ∈I, in other words, rI ⊆I for every r ∈R (3) If a ∈I and r ∈R then ar ∈I, in other words, Ir ⊆I for every r ∈R If (1) and (2) hold, I is called a left ideal of R. If (1) and (3) hold, I is called a right ideal of R. If all three properties are satisfied, I is called an ideal (or two-sided ideal) of R, a proper ideal if I ̸= R, a nontrivial ideal if I is neither R nor {0}. If f : R →S is a ring homomorphism, its kernel is kerf = {r ∈R : f(r) = 0}; exactly as in (1.3.13), f is injective if and only if kerf = {0}.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Construction of Quotient Rings Let I be a proper ideal of the ring R. Since I is
of Quotient Rings Let I be a proper ideal of the ring R. Since I is a subgroup of the additive group of R, one can form the quotient group R/I, consisting of cosets r + I, r ∈R. Define multiplication of cosets in the natural way: (r + I)(s + I) = rs + I. To show that multiplication is well-defined, suppose that r + I = r′ + I and s + I = s′ + I, so that r′ −r is an element of I, call it a, and s′ −s is an element of I, call it b.
Proof / verification strategy: Translate the element statement into ideals or quotient rings, establish both inclusions or implications, and use the strongest ring hypothesis actually available.
Proposition
Every proper ideal I is the kernel of a ring homomorphism.
Proof / verification strategy: Translate the element statement into ideals or quotient rings, establish both inclusions or implications, and use the strongest ring hypothesis actually available.
Proposition
If f : R →S is a ring homomorphism and the only ideals of R are {0} and R, then f is injective. (In particular, if R is a division ring, then R satisfies this hypothesis.)
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Definitions and Comments
If X is a nonempty subset of the ring R, then < X > will denote the ideal generated by X, that is, the smallest ideal of R that contains X.Explicitly, < X > = RXR = the collection of finite sums of the form i rixisi with ri, si ∈R and xi ∈X. To show that this is correct, verify that the finite sums of the given type form an ideal containing X. On the other hand, if J is any ideal containing X, then all finite sums i rixisi must belong to J. If R is commutative, then rxs = rsx, and one may as well drop the s.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Quick-reference relationships
Problem-solving workflow
Fix the ring hypotheses
Record commutativity, identity, zero-divisor assumptions and whether the ring is a domain, field, PID, UFD or Euclidean domain.
Translate element questions into ideal questions
Divisibility, kernels, quotients and maximality often become clearer when expressed through generated ideals.
Choose a universal construction
For quotients, fractions or polynomial evaluation, define the candidate map and prove it is well-defined.
Separate existence from uniqueness
Division, factorisation and decomposition results often require different arguments for the two directions.
Use the strongest justified structure
Do not use field division in a general ring or unique factorisation before its hypotheses have been established.
Check the result in a concrete ring
Integers, residue rings and polynomial rings provide useful sanity checks for the abstract statement.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Using cancellation in a ring that may contain zero divisors.
- Treating every irreducible element as prime without the needed domain hypothesis.
- Assuming every ideal is principal.
- Applying polynomial root counting without an integral-domain hypothesis.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 2.2 | Ideals, Homomorphisms, and Quotient Rings | 29–31 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.