What does the sentence "The Jacobson radical in matrix ring is homogeneous." mean? Is it a true or false statement?
Any way, what does "homogeneous" mean in regard of matrix theory?
Any answer or introducing a reference is appreciated!
2026-04-03 12:40:06.1775220006
Meaning of Homogeneity in matrix theory
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An ideal $I$ of an associative S-graded ring $R$ is called homogeneous if $$ I=\bigoplus_{s\in S}I_s, $$ where $I=I\cap R_S$, and $R=\oplus_{s\in S}R_s$ is the grading, with $R_sR_t\subseteq R_{st}$. In this sense, we use "homogeneous" for the Jacobson radical, see here.