mathematics

Regular local ring

Regular local ring is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Regular local ring rather than just read about it. In short: In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A {\displaystyle A} be any Noetherian local ring with unique maximal ideal m {\displaystyle {\mathfrak {m}}} , and suppose a 1 , ⋯ , a n {\displaystyle a_{1},\cdots ,a_{n}} is a minimal set of generators of m {\displays…

Key takeaways

Regular local ring belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Regular local ring to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Regular local ring from memory before moving on to harder problems.

Reference excerpt

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A {\displaystyle A} be any Noetherian local ring with unique maximal ideal m {\displaystyle {\mathfrak {m}}} , and suppose a 1 , ⋯ , a n {\displaystyle a_{1},\cdots ,a_{n}} is a minimal set of generators of m {\displaystyle {\mathfrak {m}}} . Then Krull's principal ideal theorem implies that n ≥ dim ⁡ A {\displaystyle n\geq \dim A} , and A {\displaystyle A} is regular whenever n = dim ⁡ A {\displaystyle n=\dim A} . The concept is motivated by its geometric meaning. A point x {\displaystyle x} on an algebraic variety X {\displaystyle X} is nonsingular (a smooth point) if and only if the local ring O X , x {\displaystyle {\mathcal {O}}_{X,x}} of germs at x {\displaystyle x} is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions:

Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings

Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. If A {\displaystyle A} is a Noetherian local ring with maximal ideal m {\displaystyle {\mathfrak {m}}} , then the following are equivalent definitions. A {\displaystyle A} is regular whenever:

Its Krull dimension is equal to the minimal number of generators of its maximal ideal; that is when m = ( a 1 , … , a n ) {\displaystyle {\mathfrak {m}}=(a_{1},\ldots ,a_{n})} , where n {\displaystyle n} is chosen as small as possible, and

dim ⁡ A = n {\displaystyle \dim A=n\,} . The generators { a 1 , … , a n } {\displaystyle \{a_{1},\ldots ,a_{n}\}} are then called a regular system of parameters. The dimension of its Zariski tangent space is equal to its Krull dimension; that is, when k = A / m {\displaystyle k=A/{\mathfrak {m}}} is the residue field of A {\displaystyle A} , and

dim k ⁡ m / m 2 = dim ⁡ A {\displaystyle \dim _{k}{\mathfrak {m}}/{\mathfrak {m}}^{2}=\dim A\,} . Its global dimension is finite; that is, when gl dim A := sup { pd ⁡ M ∣ M is an A -module } {\displaystyle {\mbox{gl dim }}A:=\sup\{\operatorname {pd} M\mid M{\text{ is an }}A{\text{-module}}\}} is the supremum of the projective dimensions of all A {\displaystyle A} -modules, and

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Regular local ring

Start with the simplest possible case. Write down what Regular local ring claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Regular local ring before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Regular local ring ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Regular local ring

In research: Regular local ring appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Regular local ring in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Regular local ring is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Ring theory, so understanding it makes those chapters shorter.
In everyday life: Look for Regular local ring outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Regular local ring in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Regular local ring means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Regular local ring out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Regular local ring in simple terms?

Why does Regular local ring matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Regular local ring?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Regular local ring.