Mathematicians Crack the Code for Building Better Algebraic Structures
Researchers have solved a longstanding puzzle about how mathematical ring systems preserve their fundamental properties when extended or transformed. The breakthrough could accelerate development of cryptography, coding theory, and quantum computing applications that rely on these abstract algebraic structures to function reliably at scale.
Originaltitel: The rank condition and strong rank conditions for Ore extensions
Let [Formula: see text] be a ring, [Formula: see text] a ring endomorphism, and [Formula: see text] a [Formula: see text]-derivation. We establish that the Ore extension [Formula: see text] satisfies the rank condition if and only if [Formula: see text] does. In addition, we prove analogous results for the right and left strong rank conditions. However, in the right case, the “if” part requires the hypothesis that [Formula: see text] is an automorphism, whereas, in the left case, this assumption is needed for the “only if” part. Finally, we provide a new proof of an old result of Susan Montgomery stating that a skew power series ring is directly (respectively, stably) finite if and only if its coefficient ring is directly (respectively, stably) finite.