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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated to satisfy many missing links: A __subalgebra__ is a [[subobject]] in the [[category of algebras]]. That is to say, a subalgebra of an [[associative algebra|algebra]] $A$ is a [[subset]] of $A$ that is closed under the operations of addition, multiplication, and contains all elements of the base ring. <a href="https://ncatlab.org/nlab/revision/subalgebra/1">v1</a>, <a href="https://ncatlab.org/nlab/show/subalgebra">current</a>

   Created to satisfy many missing links:

   A subalgebra is a subobject in the category of algebras.

   That is to say, a subalgebra of an algebra $A$ is a subset of $A$ that is closed under the operations of addition, multiplication, and contains all elements of the base ring.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 9th 2025
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   Author: Todd_Trimble
   Format: MarkdownItexSeems that this should be disambiguated, in view of many meanings of "algebra" (over a monad, over an endofunctor, etc.).

   Seems that this should be disambiguated, in view of many meanings of “algebra” (over a monad, over an endofunctor, etc.).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks to Dmitri for working on cleaning up and polishing the nLab, in recent days and generally, such as here with creating long-needed little infrastructure articles. It's much appreciated.

   Thanks to Dmitri for working on cleaning up and polishing the nLab, in recent days and generally, such as here with creating long-needed little infrastructure articles. It’s much appreciated.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 9th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: For other types of [[algebras]], such as [[algebras over operads]], [[algebras for endofunctors]], [[algebras over monads]], [[algebras over algebraic theories]], the notion of subalgebra is defined as a subobject in the corresponding category of algebras. <a href="https://ncatlab.org/nlab/revision/diff/subalgebra/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/subalgebra/3">v3</a>, <a href="https://ncatlab.org/nlab/show/subalgebra">current</a>

   Added:

   For other types of algebras, such as algebras over operads, algebras for endofunctors, algebras over monads, algebras over algebraic theories, the notion of subalgebra is defined as a subobject in the corresponding category of algebras.

   diff, v3, current