A definition of left R-module
Topics
About: ring
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of left R-module.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any multiplicatively-commutative ring, R, any additively-abelian group, M, together with scalar left multiplication by R, \(R \times M \rightarrow M\), such that for any \(r_1, r_2 \in R\) and \(m_1, m_2 \in M\), 1) \((r_1 r_2) m_1 = r_1 (r_2 m_1)\); 2) \(1 m_1 = m_1\); 3) \((r_1 + r_2) m_1 = r_1 m_1 + r_2 m_1\) and \(r_1 (m_1 + m_2) = r_1 m_1 + r_1 m_2\)
2: Note
It is a generalization of vector space, allowing a ring in scalar multiplication than a field.
