A left primitive ring has a faithful simple left module.
If is a simple left -module and , then every prescribed map on a finite -linearly independent subset of is induced by an element of .

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A **primitive ring** is a type of ring in which the process of "building up" the ring can be viewed as being generated by a single element, specifically, it is a ring that has a faithful module that is simple. Here is a more formal definition and some details: 1. **Definition**: A ring \( R \) is called primitive if it has no nontrivial two-sided ideals and it is simple as a module over itself.