We deal with a reduction of power series convergent in a polydisc with respect to a Gröbner basis of a polynomial ideal. The results are applied to proving that a Nash function whose graph is algebraic in a "large enough" polydisc, must be a polynomial. Moreover, we give an effective method for finding this polydisc.
We study -actions of the form , where is the dual (to ) -variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action is given. It is shown that the doubled actions have a number of nice properties, if is spherical or of complexity one.
A submodule of a -primary module of bounded order is known to be regular if and have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely that where the finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
Let be a complete multipartite graph on with and being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity is for any positive integer .
A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and (2) regularly weakly based modules over Dedekind domains.
For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.
We prove that generating relations between the elements [r] = r²-r of a commutative ring are the following: [r+s] = [r]+[s]+rs[2] and [rs] = r²[s]+s[r].