📂Abstract Algebra

Abstract Algebra in R-modules

Definitions ¹

An Abelian group $(G,+)$ with the identity element for multiplication $1 \ne 0$, and a ring $(R,+,\cdot)$ satisfy the following three conditions for the binary operation $$ \mu : R \times G \to G $$, then $\left( G, +, R, \cdot ; \mu \right)$ is called an $R$-module:

• (M1) Bi-additivity: For $\forall \alpha, \beta \in R$ and $\forall x,y \in G$, $$ \begin{align*} \mu \left( \alpha + \beta , x \right) =& \mu \left( \alpha , x \right) + \mu \left( \beta , x \right) \\ \mu \left( \alpha , x + y \right) =& \mu \left( \alpha , x \right) + \mu \left( \alpha , y \right) \end{align*} $$
• (M2): For $\forall \alpha, \beta \in R$ and $\forall x \in G$, $$ \mu \left( \alpha , \mu \left( \beta , x \right) \right) = \mu \left( \alpha \beta , x \right) $$
• (M3): For $\forall x \in G$, $$ \mu (1, x) = x $$

Here, $R$ is also referred to as the ground ring or base ring.

Explanation

Here, $\mu$ is called the scalar multiplication, and the resulting element, the scalar product, is denoted as $\mu (\alpha, x) := \alpha x$. According to this expression, the above three conditions can be represented as follows:

• (M1) Distributive law: $$ \begin{align*} \left( \alpha + \beta \right) x =& \alpha x + \beta x \\ \alpha \left( x + y \right) =& \alpha x + \alpha y \end{align*} $$
• (M2) Associative law: $$ \alpha \left( \beta x \right) = \left( \alpha \beta \right) x $$
• (M3) Identity: $$ 1 x = x $$

These might feel familiar right from the term scalar multiplication, as they have been seen often in the definition of vector spaces in linear algebra. In this sense, an $R$-module is a generalization of the $F$-vector space.

See Also

• Vector space in linear algebra
• Vector space in abstract algebra

The $F$-vector space discussed in the documents below is essentially no different from the above-mentioned vector spaces. However, the perspectives differ; the vector space in linear algebra is an abstraction of the intuitive Euclidean space, while the vector space in abstract algebra is truly regarded as ‘algebra’.

Conversely, the $R$-module generalizes the scalar field $F$ of the $F$-vector space into the scalar ring $R$, thus demonstrating its significance. From the perspective of group $G$, the addition of the ring $R$ and the new operation $\mu$ could be deemed as forming an additive group.

• $R$-modules in abstract algebra
• $F$-Vector space in abstract algebra