Ring whose every simple singular module is faithful.

Question

Is there any ring (not necessary commutative), other than simple ring, whose every simple singular right $R-$module is faithful?

A $R-module$ $M$ is called faithful if $ann_r(M)=0.$

Answer 1

All simple right $R$ modules of a semisimple ring $R$ are not singular, and in that case it would vacuously satisfy your condition.

So, for example, $F_2\times F_2$ ($F_2$ the field of two elements) satisfies the condition and is not simple.

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If you ask for at least one faithful simple singular module to exist, then you'd need examples of right primitive rings that aren't simple, which are harder to come by. In fact, both of those examples have nonzero right socles, and so they cannot have faithful singular right modules.