Definition
In this definition the rings are assumed to be associative, but not necessarily commutative, or to have an identity. Also, modules are assumed to be left modules. The modifications needed in the case of right modules are straightforward.
Let be a homomorphism between two rings, and let be a module over . Consider the tensor product, where is regarded as a right -module via . Since is also a left module over itself, and the two actions commute, that is for, (in a more formal language, is a -bimodule), inherits a left action of . It is given by for and . This module is said to be obtained from through extension of scalars.
Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an bimodule with an R-module is an S-module.
Read more about this topic: Extension Of Scalars
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