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

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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 ?

#### Solutions Collecting From Web of "What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?"

Paul Pollack proved in this paper 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 $1965$.