Localisation and Primary Decomposition
Handbook guide to localisation and primary decomposition with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Localization
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.
Primary Decomposition
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.
Geometric Motivation Suppose that V is an irreducible variety, so that I(V ) is
Geometric Motivation Suppose that V is an irreducible variety, so that I(V ) is a prime ideal. A polynomial g will belong to I(V ) if and only if it vanishes on V . If we are studying rational functions f/g in the neighborhood of a point x ∈V , we must have g(x) ̸= 0. It is very convenient to have every polynomial g /∈I(V ) available as a legal object, even though g may vanish at some points of V .
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Notation Recalling the setup of Section 2.8, let S be a multiplicative subset of the
Notation Recalling the setup of Section 2.8, let S be a multiplicative subset of the ring R, and S−1R the ring of fractions of R by S. Let h be the natural homomorphism of R into S−1R, given by h(a) = a/1. If X is any subset of R, define S−1X = {x/s : x ∈X, s ∈ S}. We will be especially interested in such a set when X is an ideal.
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.
Lemma
If I is an ideal of R, then S−1I is an ideal of S−1R. If J is another ideal of R, then (i) S−1(I + J) = S−1I + S−1J; (ii) S−1(IJ) = (S−1I)(S−1J); (iii)S−1(I ∩J) = S−1I ∩S−1J; (iv) S−1I is a proper ideal iffS ∩I = ∅.
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.
Lemma
If J is an ideal of S−1R and I = h−1(J), then I is an ideal of R and S−1I = J.
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.
Lemma
If I is any ideal of R, then I ⊆h−1(S−1I), with equality if I is prime and disjoint from S.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Lemma
If I is a prime ideal of R disjoint from S, then S−1I is a prime ideal of S−1R.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Theorem There is a one-to-one correspondence between prime ideals P of R that
There is a one-to-one correspondence between prime ideals P of R that are disjoint from S and prime ideals Q of S−1R, given by P →S−1P and Q →h−1(Q).
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Definitions and Comments
If P is a prime ideal of R, then S = R \ P is a multiplicative set. In this case, we write R(P) for S−1R, and call it the localization of R at P. (The usual notation is RP , but it’s easier to read without subscripts.) If I is an ideal of R, we write I(P) for S−1I. We are going to show that R(P) is a local ring, that is, a ring with a unique maximal ideal.
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.
Proposition
For a ring R, the following conditions are equivalent. (i) R is a local ring; (ii) There is a proper ideal I of R that contains all nonunits of R; (iii) The set of nonunits of R is an ideal.
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.
Localization of Modules If M is an R-module and S a multiplicative subset of
Localization of Modules If M is an R-module and S a multiplicative subset of R, one can essentially repeat the construction of Section 2.8 to form the localization S−1M of M by S, and thereby divide elements of M by elements of S. If x, y ∈M and s, t ∈S, we call (x, s) and (y, t) equivalent if for some u ∈S, u(tx −sy) = 0. The equivalence class of (x, s) is denoted by x/s, and addition is defined by x s + y t = tx + sy st . If a/s ∈S−1R and x/t ∈S−1M, define a s x t = ax st .
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definition
The ideal Q in the ring R is primary if Q is proper and whenever a product ab belongs to Q, either a ∈Q or bn ∈Q for some positive integer n. [The condition on b is equivalent to b ∈√Q.] An equivalent statement is that R/Q ̸= 0 and whenever (a+Q)(b+Q) = 0 in R/Q, either a+Q = 0 or (b+Q)n = 0 for some positive integer n. This says that if b + Q is a zero-divisor in R/Q, then it is nilpotent, that is, some power of b + Q is 0. It follows from the definition that every prime ideal is primary. Also, if Q is primary, then √Q is the smallest prime ideal containing Q. [Since √Q is the intersection of all prime ideals containing Q (Section 8.3, Problem 2), it suffices to show that √Q is prime.
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.
Examples
1. In Z, the primary ideals are {0} and (pr), where p is prime. In Z6, 2 and 3 are zerodivisors that are not nilpotent, and a similar situation will occur in Zm whenever more than one prime appears in the factorization of m. 2. Let R = k[X, Y ] where k is any field, and take Q = (X, Y 3), the ideal generated by X and Y 3.
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.
Lemma
If P is a prime ideal, then for every positive integer n, √ P n = P.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Definition
The nilradical N(R) of a ring R is the set of nilpotent elements of R, that is, {x ∈R : xn = 0 for some positive integer n}. Thus N(R) is the radical of the zero ideal, which is the intersection of all prime ideals of R.
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.
Proposition
If the radical of the ideal Q is maximal, then Q is primary.
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.
Corollary
If M is a maximal ideal, then M n is M-primary for all n = 1, 2, . . ..
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 Q is a finite intersection of P-primary ideals Qi, i = 1, . . . , n, then Q is P-primary.
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.
Lemma
Call an ideal I irreducible if for any ideals J and K, I = J ∩K implies that I = J or I = K. If R is Noetherian, then every ideal of R is a finite intersection of irreducible ideals.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Primary Decomposition Theorem Every proper ideal in a Noetherian ring R has
Primary Decomposition Theorem Every proper ideal in a Noetherian ring R has a primary decomposition. (One can drop the word “proper” if we regard R as the intersection of the empty collection of primary ideals.)
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.
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 supplied worked solutions for this section repeatedly test subgroup, ideal, polynomial, module, exact, prime, maximal, radical. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Reversing the inclusion direction between ideals and zero sets.
- Replacing an ideal by its variety and expecting to recover the same ideal rather than its radical.
- Localising without checking that the chosen set is multiplicatively closed and avoids zero.
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 |
|---|---|---|
| 8.5 | Localization | 165–167 |
| 8.6 | Primary Decomposition | 168–169 |
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.