Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. modules over a commutative ring) it can be related to things like one-dimensionality. In the general case, I don't know whether algebras embeddable into a projective algebra may be easily characterized by their category-theoretic properties.

> Are there results about this? How to characterize subobjects of projective objects?

Most likely the question whether the class of projective algebras in a variety is closed under ultraproducts is even harder, and condition might be very restrictive for all that I know. I have vague feeling that ultraproducts of projectives must be close to being just flat (although I do not know what flatness means in general varieties). I also guess this question must be thoroughly investigated. So,

> Can one characterize ultraproducts of projectives in category-theoretic terms? When are they again projective? Where to read about this?

One more vague feeling - it might be that the similar question about **finitely generated** projectives yields more natural and more easily described answer. So, what are ultraproducts of f.g. projectives, and when are they again projective?