What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?

Question

All:

what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ?

In other word, for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, etc.

What does "even" element mean ? What does "prime" element mean ?

Can every "even" element for those algebraic structures be partitioned into two "prime" elements ?

Has Goldbach Conjecture been proved for other algebraic structures ?

Answer 1

Paul Pollack proved in this paper¹ the following version of Goldbach for polynomial rings:

Theorem (Pollack): let $R$ be a Noetherian ring with infinitely many maximal ideals. Then every polynomial in $R[x]$ of degree $n\ge 1$ can be written as the sum of two irreducible polynomials of degree $n$.

For $R=\mathbb{Z}$ this has been proved by David Hayes in $1965$.

¹P. Pollack: On Polynomial Rings with a Goldbach Property jstor, author's website (Wayback Machine).