… A and I an ideal of R. Suppose I contains a monic polynomial and #(I/I 2) >= dim (R/I) + 2.
Then #(I) = p(I/I2). In the proof of Theorem 1.2 we shall actually extend the ideal I to the Laurent …

… If R is a ring and I is an ideal in R, then we shall use ht(1) and dim(I) to denote the height of I
… We may as well cite that the requirement on the residue field to be infinite is necessary. For …

… GENERATION OF MAXIMAL IDEALS IN POLYNOMIAL … in this case without any regularity
assumption on R. The … of the theorem we proceed by induction on hl(M). Suppose ht(Af…

… As applications, we prove some interesting results on set-theoretic generation of ideals in
a … We prove the result using induction on the number of variables appearing in a minimal …

… This depends very much on the types of rings and the nature of ideals therein. We shall
use the following theorem, which is essentially due to Mohan Kumar[11]. THEOREM B. …

… This chapter explains the fact that improvement of the effect of a typical photon on the electron
… The first and most obvious attack on this problem is to use thermodynamics, and we offer …

… 3.2 Empirical and Theoretical Investigation of Data-centric Efficient Learning: a Case
Study To better understand the ideal data properties of training on DX and DY , we consider …

… Whenever in the Buchberger algorithm we perform a computation on the first component,
we perform the same operation on the second component. In this way, any pair (g;v) that we …

… on the Prüfer domain R also imply that a power of each finitely generated maximal ideal
of R is principal. This result naturally raises the question whether the same conditions on R …

… energy and entropy generation caused by heat transfer between the propelling fluid and the
… generator. By minimizing S σ inevitable losses of power and limits on the work generation …