Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jaco...

We prove that all trace $1$, $2\times 2$ invertible matrices over $\mathbb{Z}$ are nil-clean and, up to similarity, there only two $\mathbb{Z}$.

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...