Twisted Rings, An Introduction

Introduction

Let R be a ring, and adjoin x, but let x*c = σ(c)*x, where c is a coefficient drawn from R, and σ is a ring endomorphism of R. This extension is called a twisted ring.

A polynomial is in normal form if its coefficients appear on the left. Multiply any three terms together and verify associativity. Use the fact that σ is a ring homomorphism.

ax^k * bx^l * cx^m = aσ^k(b)σ^k+l(c)x^k+l+m

Apply the above to all terms in the expanded product of three polynomials, and multiplication becomes associative.

Verify the distributive property, on the left and on the right. This works because σ(a+b) = σ(a) + σ(b). The other properties are straightforward. Therefore the twisted extension is really a ring.

If σ is the identity map, R commutes with x, giving the standard polynomials R[x].

Since ring homomorphisms map 1 to 1 (the default assumption on this website), σ is assumed to be nontrivial. If σ was trivial then x = x*1 = 0*x = 0, and we are back to R.

If σ is not the identity map, the ring is noncommutative. Of course, R could be noncommutative to begin with.

When R has characteristic p, σ can be set to the frobenius homomorphism c^p.

Another extension is the twisted series, which is based on the formal power series R[[x]]. Once again xc = σ(c)x. If there is any ambiguity, I will use the term "twisted polynomials" for finite polynomials, and "twisted series" for infinite series.

Twisted laurent series are also possible, wherein each series can have finitely many terms with negative exponents. This assumes σ is an invertible automorphism, so that constants can be pulled past negative powers of x. These rings are not fundamentally different from the twisted series, so I won't spend a lot of time on them. Just remember that a theorem regarding twisted series can usually be generalized to twisted laurent series.