Let be a ring and let be a left -module. Recall the following definitions:

1) is called **faithful** if implies for any In other words, is called faithful if

2) is called **simple** if and are the only left submodules of

Faithful and simple right -modules are defined analogously.

**Definition.** A ring is called left (resp., right) **primitive** if there exists a left (resp., right) -module which is both faithful and simple.

**Remark.** We will show later that left and right primitivity are not equivalent. From now on, I will only consider left primitive rings. If a statement is true for left but not for right, I will mention that.

**Example 1**. If is a ring and is a left simple -module, then is a left primitive ring. This is clear because would be a faithful simple left -module.

**Example 2**. Every simple ring is left primitive. That’s because we can choose a maximal left ideal of and then would be a faithful simple left -module. The reason that is faithful is that is a two-sided ideal contained in and therefore because is simple. One special case of this example is the ring of matrices with entries from a division ring If is an infinite dimensional vector space over a field , then is an example of a left primitive ring which is not simple [see Example 4 and this post].

**Example 3**. If is a left primitive ring and an idempotent, then is left primitive: let be a faithful simple -module. The claim is that is a faithful simple left -module. Note that because and is faithful. Clearly is a left -module. To see why it’s faithful, let with Then So because is faithful. To prove that is a simple -module let We need to show that Well, since for some we have Thus

**Example 4**. Let be a division ring and let be a right vector space over Then is a left primitive ring. Here is why: is a left -module if we define for all and It is clear that is faithful as a left -module. To see why it is simple, let with Let be a basis for over such that for some Define by Then So, we’ve proved that which shows that is a simple -module. If then which we already showed its primitivity in Example 2. Note that if was a “left” vector space over with then would be isomorphic to the ring rather than the ring

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