Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Algebra homomorphism
Summary
In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function such that, for all k in K and x, y in A, one has
The first two conditions say that F is a K-linear map, and the last condition says that F preserves the algebra multiplication. So, if the algebras are associative, F is a rng homomorphism, and, if the algebras are rings and F preserves the identity, it is a ring homomorphism.
If F admits an inverse homomorphism, or equivalently if it is bijective, F is said to be an isomorphism between A and B.
If A and B are two unital algebras, then an algebra homomorphism is said to be unital if it maps the unity of A to the unity of B. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.
A unital algebra homomorphism is a (unital) ring homomorphism.
Every ring is a -algebra since there always exists a unique homomorphism . See Associative algebra#Examples for the explanation.
Any homomorphism of commutative rings gives the structure of a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map is a homomorphism of commutative rings. It is straightforward to deduce that the of the commutative rings over R is the same as the category of commutative -algebras.
If A is a subalgebra of B, then for every invertible b in B the function that takes every a in A to b−1 a b is an algebra homomorphism (in case , this is called an inner automorphism of B). If A is also simple and B is a central simple algebra, then every homomorphism from A to B is given in this way by some b in B; this is the Skolem–Noether theorem.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (8)
MATH-410: Riemann surfaces
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
MATH-506: Topology IV.b - cohomology rings
Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a
COM-502: Dynamical system theory for engineers
Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the quali