Finite algebra¶
An R-algebra that is finitely generated as an R-module, a stronger finiteness condition than finite generation as an algebra.
Core Idea¶
An (R)-algebra (A) is finite when it is finitely generated as an (R)-module through a specified structure map (R o A). A finite set must span every element by (R)-linear combination, a stronger requirement than having finitely many algebra generators. The condition belongs to the specified base map. The condition belongs to the specified base map.
Scope of Application¶
The condition applies to ring extensions and geometric morphisms whose coordinate algebras are module-finite over the base. The condition applies to relative ring extensions and to affine or scheme morphisms defined locally by module-finite coordinate algebras.
- Integral extensions. Module-finite algebras are integral over their base.
- Affine geometry. Finite coordinate-ring maps define finite affine morphisms.
- Scheme theory. Local module finiteness extends the construction beyond affine schemes.
- Number theory. Orders and rings of integers are studied as finite modules over base rings.
- Representation and invariant theory. Finite module structure controls algebraic dependence and size.
Clarity¶
State the structure map and whether rings are assumed commutative and unital. Exhibit module generators or invoke a theorem that provides them. Keep finite, finite type, finitely presented, integral, and finite-dimensional over a field distinct, noting hypotheses for implications. The closest near miss sets the boundary: Finite type is the closest near miss: A is generated by finitely many algebra elements, but their arbitrarily high products can prevent finite module generation.
Manages Complexity¶
A finite spanning set reduces an entire algebra and all powers of its elements to bounded module data. This gives strong control under localization and supports passage between algebraic ring maps and geometric finite morphisms. The central finite algebra generators–finite module span tradeoff is this: Products of a small algebra-generating set can still create infinitely many independent module terms. A second relative property–intrinsic language tension matters because The adjective sounds absolute although the base map is essential.
Abstract Reasoning¶
Use three linked moves: fix the base homomorphism R→A and induced module action; propose finitely many module generators and show they span every element of A; alternatively prove integrality plus finite type under hypotheses that imply module finiteness. As a collapse test, the case exits when no finite R-module spanning set exists for the declared base map. A fourth check is to check the property after localization when working geometrically. A final check is to do not infer module finiteness solely from finite fibers or algebra generators.
Knowledge Transfer¶
Finite-generation patterns transfer to groups, modules, and algebras, but the finite-algebra label specifically uses module generation relative to a base ring. The geometric translation is literal only through the coordinate-ring or sheaf condition. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. A finite set spans the target under allowed operations. Infinite content is controlled by bounded data.
Relationships to Other Abstractions¶
Current abstraction Finite algebra Domain-specific
Parents (1) — more general patterns this builds on
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Finite algebra is a kind of Algebra over a Ring Domain-specific
A finite algebra is an algebra over R whose underlying R-module is finitely generated.
Hierarchy path (1) — routes to 1 parentless root
- Finite algebra → Algebra over a Ring → Linearity
Neighborhood in Abstraction Space¶
Finite algebra sits in a sparse region of the domain-specific corpus (62nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Ring & Module Structure Theory (14 abstractions)
Nearest neighbors
- Group Ring — 0.85
- Associative algebra — 0.85
- Ring — 0.85
- Serial Module — 0.85
- Burnside category — 0.84
Computed from structural-signature embeddings · 2026-10-08