In algebra, Posner's theorem states that given a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
polynomial identity algebra In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X'N'' such th ...
''A'' with center ''Z'', the ring
is a
central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simp ...
over
, the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''Z''.
It is named after
Ed Posner
Edward Charles "Ed" Posner (August 10, 1933 – June 15, 1993) was an American information theory, information theorist and neural network researcher who became chief technologist at the Jet Propulsion Laboratory and founded the Conference on Neura ...
.
References
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Edward C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), pp. 180–183.
{{algebra-stub
Theorems in ring theory