Can a projective module be singular?

Question

Let $R$ be a ring with unity, and $M$ be a cyclic projective $R$-module. I know that $M$ can not be singular.
My question is, if we remove the cyclic condition, will the conclusion hold?

Answer 1

Good question. I remembered seeing versions of this before like this:

If $N$ is essential in $M$, then $M/N$ is singular. The converse is true if $M$ is free.

and a variation on that

If $M=R/T$ is a cyclic, projective, singular module, it is zero (since the result above says $T$ is essential, and if $R/T$ is projective $T$ splits out of $R$ as a summand, so $T=R$.)

Now it seems by the same logic that

If $M=F/T$ is a projective singular module, it is zero (since $T$ is a summand of $F$ that is essential, $T=F$.)

Strangely enough when I checked, I saw the first two versions above in many sources, and did not find the third one in many places, except by a few places by T. H. Loggie who says it is well-known, and a paper by D. Zhou (who seems to use it without explanation.)

So all is well if you can show the direction of the first version above:

If $M$ is free and $M/N$ is singular, then $N$ is essential in $M$.

I believe this is usually listed as an exercise but if you need help I can go into details.