A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Allegory, It Happens: A Multi-Perspective Case Study of The Lord of the Rings

Allegory is not obsolete as Samuel Coleridge and Johann Wolfgang von Goethe have claimed. It is alive and well and has transformed from a restrictive concept to a concept that is flexible and can form to meet the needs of the author or reader. The most efficient way to evidence this is by making a case study of it with a suitable work that will allow us to perceive its plasticity. This essay uses J.R.R. Tolkien’s The Lord of the Rings as a multi-perspective case study of the concept of allegory; the size and complexity of the narrative make it a suitable choice. My aim is to illustrate the plasticity of allegory as a concept and illuminate some of the possibilities and pitfalls of allegory and allegoresis. As to whether The Lord of the Rings can be treated as an allegory, it will be examined from three different perspectives: as a purely writerly process, a middle ground of writer and reader and as a purely readerly process. The Lord of the Rings will then be compared to a series of concepts of allegorical theory such as Plato’s classical “The Ring of Gyges”, William Langland’s classic The Vision of William Concerning Piers the Plowman and contemporary allegories of racism and homoeroticism to demonstrate just how adaptable this concept is. The position of this essay is that the concept of allegory has changed over time since its conception and become more malleable. This poses certain dangers as allegory has become an all-round tool for anyone to do anything that has few limitations and has lost its early rigid form and now favours an almost anything goes approach.