A commutative semisimple ring is a finite direct product of fields.

A finite or countable product of continua is a continuum.

Conversely, any finite product of integrally closed domains is normal.

In fact any category with finite products can be given a monoidal structure.

Principal right ideal rings are closed under finite direct products.

Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.

A is a finite product of commutative Artinian local rings.

This can be rewritten as a finite product as follows.

This is easy to show for finite products, while the general statement is equivalent to the axiom of choice.

The matrix A can be expressed as a finite product of elementary matrices.