Inheritance of generalized t-fineness by full matrix rings

Establish whether, for every generalized t-fine ring R and every positive integer n, the full matrix ring M_n(R) is generalized t-fine.

Background

The paper studies when the decomposition property R\setminus J(R)=T(R)+(R), where T(R) denotes torsion units and (R) denotes nilpotent elements, is preserved under passage to full matrix rings. It proves preservation for weakly 2-primal generalized t-fine rings and for algebras over fields with at least three elements, but does not settle the unrestricted case.

The unresolved issue is therefore whether generalized t-fineness is inherited by M_n(R) without additional hypotheses on R, such as weak 2-primality or an algebra structure over a sufficiently large field.

References

However, we are currently unable to establish the general result that being generalized $t$-fine is inherited by full matrix rings.

— Generalized $t$-Fine and Quasi $t$-Fine Rings  (2609.19882 - Bien et al., 17 Sep 2026) in Section 2, subsection “Matrix Rings,” immediately before Theorem 2.?? (the theorem labeled \ref{matrix1})

Although $\mathbb M_n(\mathbb F_2)$ is $t$-fine by Theorem~6, we are presently unaware of what happens to the generalized $t$-fine structure of $\mathbb M_n(R)$ when $R$ is an algebra over $\mathbb F_2$.

— Generalized $t$-Fine and Quasi $t$-Fine Rings  (2609.19882 - Bien et al., 17 Sep 2026) in Section 2, subsection “Matrix Rings,” immediately after Theorem 2.?? (the theorem labeled \ref{matrix2})