---
created: 2023-04-14T16:47
updated: 2023-04-14T16:47
---

---
date: 2022-02-23 18:45
modification date: Tuesday 22nd March 2022 21:37:40
title: algebra over a ring
aliases: [algebra over a ring, algebra, algebra structure, k-algebra, algebra morphism]
---

---

- Tags 
	- #AG/schemes 
- Refs:
	- #todo/add-references
- Links:
	- [coalgebra](Unsorted/coalgebra.md)
	- [associated algebra](associated%20algebra)
	- [symmetric algebra](symmetric%20algebra)
	- [polynomial algebra](polynomial%20algebra)
	- [exterior algebra](exterior%20algebra)
	- [graded algebra](graded%20algebra)
	- [DGA](Unsorted/DGA.md)
	- [Lie algebra](Unsorted/Lie%20algebra.md)
	- [Clifford algebra](Unsorted/Clifford%20algebra.md)
	- [Hopf algebra](Hopf%20algebra.md)
	- [quadratic form](Unsorted/quadratic%20form.md)

---

# algebra over a ring

![[algebra over a ring 2023-04-14 16.46.21.excalidraw]]

Definition: an $R\dash$algebra is a unital ring $A$ with a ring morphism $R\mapsvia{f} A$ with $1_R \mapsto 1_A$ such that $\im f \subseteq Z(A)$.

Equivalently, $A$ is an $R\dash$module with a compatible ring structure, i.e. a product $(a_1, a_2) \mapsto a_1 . a_2$ such that $r(a_1.a_2) = (ra_1).a_2 = a_1.(ra_2)$,

An $A$-algebra $B$ is the same data as a ring map $A \rightarrow B$.
Ways to remember which direction this should go: 

- $\CRing \cong \CRing\coslice{\ZZ}\cong\zalg$ where $\ZZ\to \RR$ is determined by $1\mapsto 1_R$.
- $M \da \Mat_{n\times n}(R)$ is an $R\dash$algebra where $f: R\to M$ by $r\mapsto \diag(r,r,c\dots, r) \in Z(M)$.
- Field extensions $L/K$ admit ring morphisms $K\to L$ making $L$ a $K\dash$algebra.