# Ring detail

## Name: Nakayama's quasi-Frobenius ring that isn't Frobenius

Description: Let $F_2$ be the field of two elements, and consider the subring of matrices of the form $\begin{bmatrix}a&b&p&0&0&0\\c&d&q&0&0&0\\0&0&r&0&0&0\\0&0&0&r&s&t\\0&0&0&0&a&b\\0&0&0&0&c&d\end{bmatrix}$ with entries in $F_2$.

Notes: Jacobson radical is the subset with $a=b=c=d=r=0$.

Keywords matrix ring subring

Reference(s):

• T. Nakayama. On Frobeniusean algebras. I. (1939) @ pp 61-633
• T.-Y. Lam. Lectures on modules and rings. (2012) @ Example 16.19(5) p 429

Legend
• = has the property
• = does not have the property
• = information not in database
Name Measure
cardinality 512
Krull dimension (classical) 0
Name Description
Jacobson radical The elements with $a=b=c=d=r=0$.