Radicals, String Representable

String Representable

Let R be Z adjoin arbitrarily many indeterminants t₁ t₂ t₃ etc, which do not commute with each other, and let K be a kernal ideal, so that R/K is an arbitrary noncommutative ring. Assume a polynomial is in K iff every term in said polynomial is in K. Thus 3t₁+4t₂t₃ in K means 3t₁ and 4t₂t₃ lie in K. In this case the kernel is "string representable", i.e. generated by strings. The relators in the presentation are strings, not polynomials. Similarly the ring R/K is called string representable.

Left Nil Ideal Generates a Nil Ideal

Every left nil ideal H in R/K generates a nil ideal. Thus R/K is kotherian. Review the previous section for a description of kotherian rings.

Let w be an arbitrary element in the ideal generated by H. Thus w is a linear combination of elements of H, with coefficients of R/K on the right. We'll use elements from R, which define cosets of K. Thus w is the finite sum of g_ic_i, where each g_i and c_i comes from R, and each g_i mod K lies in H, which is a left nil ideal in R/K.

Let 1, g_i, and c_ig_i produce a finitely generated subring U inside R. Let s be the sum of these generators (other than 1). Note that s lies in H, and is nilpotent. Thus sⁿ is in K.

Expand sⁿ into a sum of products, and each possible product of length n appears. Since K is string representable, each product of length n lies in K.

Expand wⁿ into a sum of products. Each product is a string of length n, using the generators of U, times something from R on the right. Each product lies in K, and drops to 0, hence wⁿ = 0 in R/K. Therefore H generates a nil ideal.