Deleted Question on Semi-Simple and Projective but not Injective Module

A backup of a deleted PSQ : OP : irfanmat It has a detailed answer by Atticus Stonestrom. It’s pity that his post got deleted. As there’s no reason for undeletion, I’m posting it here so as to preserve the contents. Question body Is there a semi-simple and projective but not injective module? I will be glad if you help. Response(s) In the non-commutative case, the answer is yes. Consider $R$ the ring of upper triangular $2\times 2$ matrices over a field $F$, and denote by $e_{ij}$ the element of $R$ with the $ij$-th entry equal to $1$ and all other entries equal to $0$. [Read More]