Ring and field methods organise addition, multiplication, divisibility, ideals and polynomial equations. The main task is to identify which ring properties are available before borrowing intuition from the integers or from fields. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Learning pathRing and Field Theory
LevelAdvanced
FormatHandbook guide
Read time11 min
Executive summary
This chapter develops ring homomorphisms, ideals and kernel methods as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
State the ambient ring or field and whether multiplication is commutative.
Identify units, zero divisors, ideals and the relevant quotient or extension.
For divisibility questions, distinguish irreducible elements from prime elements unless the setting makes them equivalent.
For polynomial questions, record the coefficient ring and the degree assumptions.
Use kernels and ideals to control quotient constructions and induced maps.
Verify that every division, cancellation or inverse used is valid in the stated algebraic structure.
Core definitions
Definition
If A and R are (commutative) rings, a (ring) homomorphism is a function f : A →R such that (i) f (1) = 1; (ii) f (a + a′) = f (a) + f (a′) for all a, a′ ∈A; (iii) f (aa′) = f (a) f (a′) for all a, a′ ∈A. A homomorphism that is also a bijection is called an isomorphism. Commutative rings A and R are called isomorphic, denoted by A ∼= R, if there is an isomorphism f : A →R.
Definition
If f : A →R is a ring homomorphism, then its kernel is ker f = {a ∈A with f (a) = 0}, and its image is im f = {r ∈R : r = f (a) for some a ∈R}. Notice that if we forget their multiplications, then the rings A and R are additive abelian groups and these definitions coincide with the group-theoretic ones. Let k be a commutative ring, let a ∈k, and, as in Example 3.46(iv), consider the evaluation homomorphism ea : k[x] →k sending f (x) ↦f (a). Now ea is always surjective, for if b ∈k, then b = ea( f ), where f (x) = x −a + b. By definition, ker ea consists of all those polynomials g(x) for which g(a) = 0; that is, ker ea consists of all the polynomials in k[x] having a as a root. The kernel of a group homomorphism is not merely a subgroup; it is a normal subgroup; that is, it is also closed under conjugation by any element in the ambient group.
Definition
An ideal in a commutative ring R is a subset I of R such that (i) 0 ∈I; (ii) if a, b ∈I, then a + b ∈I;9 (iii) if a ∈I and r ∈R, then ra ∈I. The ring R itself and the subset consisting of 0 alone, which we denote by {0}, are always ideals in a commutative ring R. An ideal I ̸= R is called a proper ideal.
Definition
A domain R is a principal ideal domain if every ideal in R is a principal ideal. This name is often abbreviated to PID.
Definition
An element δ in a commutative ring R is a greatest common divisor, gcd, of elements α, β ∈R if (i) δ is a common divisor of α and β; (ii) if γ is any common divisor of α and β, then γ | δ. In the special case R = Z, we force uniqueness of the gcd by requiring it to be positive; if R = k[x], where k is a field, then we force uniqueness of the gcd by further requiring it to be monic. Remark. Let R be a PID and let π, α ∈R with π irreducible. A gcd δ of π and α is, in particular, a divisor of π. Hence, π = δε, and irreducibility of π forces either δ or ε to be a unit. Now α = δβ. If δ is not a unit, then ε is a unit, and so α = δβ = πε−1β; that is, π | α. We conclude that if π ∤α, then δ is a unit; that is, 1 is a gcd of π and α.
Definition
If f and g are elements in a commutative ring R, then a common multiple is an element m ∈R with f | m and g | m. If f and g in R are not both 0, define their least common multiple, abbreviated lcm, to be a common multiple c of them with c | m for every common multiple m. If f = 0 = g, define their lcm = 0. The lcm of f and g is often denoted by [ f, g]. In the special case R = Z, we force uniqueness of the lcm by requiring it to be positive; if R = k[x], where k is a field, then we force uniqueness of the lcm by further requiring it to be monic.
Definition
A division-norm rings is a domain R that is equipped with a function ∂: R −{0} →N, called a degree function, such that (i) ∂( f ) ≤∂( f g) for all f , g ∈R with f , g ̸= 0; (ii) for all f , g ∈R with f ̸= 0, there exist q, r ∈R with g = q f + r, where either r = 0 or ∂(r) < ∂( f ). Note that if R has a degree function ∂that is identically 0, then condition (ii) forces r = 0 always; taking g = 1 shows that R is a field in this case.
Principal results and structural facts
Key result
If f : A →R is a ring homomorphism, then, for all a ∈A, (i) f (an) = f (a)n for all n ≥0; (ii) if a is a unit, then f (a) is a unit and f (a−1) = f (a)−1; in fact, if a is a unit, then f (a−n) = f (a)−n for all n ≥1; (iii) if f : A →R is a ring homomorphism, then f (U(A)) ≤U(R), where U(A) is the group of units of A; if f is an isomorphism, then U(A) ∼= U(R).
Key result
If R and S are commutative rings and ϕ : R →S is a ring homomorphism, then there is a ring homomorphism ϕ∗: R[x] →S[x] given by ϕ∗: r0 + r1x + r2x2 + · · · ↦ϕ(r0) + ϕ(r1)x + ϕ(r2)x2 + · · · .
Key result
If f : A →R is a ring homomorphism, then ker f is an ideal in A and im f is a subring of R. Moreover, if A and R are not zero rings, then ker f is a proper ideal.
Key result
If f : k →R is a ring homomorphism, where k is a field and R is not the zero ring, then f is an injection.
Key result
If k is a field, then every ideal I in k[x] is a principal ideal. Moreover, if I ̸= {0}, there is a monic polynomial that generates I.
Key result
Let R be a PID. (i) Every α, β ∈R has a gcd, δ, which is a linear combination of α and β: δ = σα + τβ, where σ, τ ∈R. (ii) If an irreducible element π ∈R divides a product αβ, then either π | α or π | β.
Source-grounded examples
Worked source example
(i) Let R be a domain and let F = Frac(R) denote its fraction field. In Theorem 3.13 we said that R is a subring of F, but that is not the truth; R is not even a subset of F. We did find a subring R′ of F, however, that has a very strong resemblance to R, namely, R′ = {[a, 1] : a ∈R} ⊆F. The function f : R →R′, given by f (a) = [a, 1], is easily seen to be an isomorphism. (ii) When an element in a commutative ring R was “identified” with a constant polynomial [in the proof of Lemma 3.16(iii)], that is, r was identified with (r, 0, 0, . . . ), we implied that R is a subring of R[x]. The subset R′ = {(r, 0, 0, . . . . . ), is an isomorphism. (iii) If S is a subring of a commutative ring R, then the inclusion i : S →R is a ring homomorphism because we have insisted that the identity 1 of R lies in S.
Worked source example
(i) The integers Z is a division-norm rings with degree function ∂(m) = |m|. In Z, we have ∂(mn) = |mn| = |m||n| = ∂(m)∂(n). (ii) When k is a field, the domain k[x] is a division-norm rings with degree function the usual degree of a nonzero polynomial. In k[x], we have ∂( f g) = deg( f g) = deg( f ) + deg(g) = ∂( f ) + ∂(g).
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
Failure mode
Control
Dividing by a nonunit.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Assuming factorisation is unique in an arbitrary domain.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Treating an irreducible element as prime without the required hypotheses.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Forgetting that polynomial behaviour depends on the coefficient ring.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Forming a quotient by a subset that is not an ideal.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Verification checklist
The ambient set, ring, field, group, module or category has been stated.
Every operation and map used is well-defined in that setting.
The hypotheses of each structural result have been checked before use.
Representatives, coordinates or generators have not been confused with the underlying object.
Existence and uniqueness have been separated where both matter.
The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about ring homomorphisms, ideals and kernel methods?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.
Connections within the handbook
Source basis: supplied advanced algebra reference. Source-identifying authorship, publisher information, acknowledgements and biographical material are intentionally omitted. Mathematical terminology and results are retained in handbook form.
