Non-commutative Ring Learning with Errors from Cyclic Algebras

Directed prime graph of non-commutative ring

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

Non-commutative Gröbner Bases for Commutative Algebras

An ideal I in the free associative algebra k〈X1, . . . ,Xn〉 over a field k is shown to have a finite Gröbner basis if the algebra defined by I is commutative; in characteristic 0 and generic coordinates the Gröbner basis may even be constructed by lifting a commutative Gröbner basis and adding commutators.

Non-cyclic Algebras with Pure Maximal Subfields*

One of the most elementary consequences of the assumption that a normal division algebra A is cyclic of degree n over its centrum K is that A contains a quantity j whose minimum equation is co=7 in K. In 1933 I conjectured the truth of the converse proposition. The proof is easily reducible f to the case where n is a power p of a prime p. Let a be the characteristic of K. I succeeded in proving...

Gelfand theory for non-commutative Banach algebras

Let A be a Banach algebra. We call a pair (G,A) a Gelfand theory for A if the following axioms are satisfied: (G 1) A is a C∗-algebra, and G : A → A is a homomorphism; (G 2) the assignment L 7→ G−1(L) is a bijection between the sets of maximal modular left ideals of A and A, respectively; (G 3) for each maximal modular left ideal L of A, the linear map GL : A/G−1(L) → A/L induced by G has dense...

Dimension Computations in Non-Commutative Associative Algebras