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Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the ...
m is a minimal prime over (x 1, ..., x d). The radical of (x 1, ..., x d) is m. Some power of m is contained in (x 1, ..., x d). (x 1, ..., x d) is m-primary. Every local Noetherian ring admits a system of parameters. [1] It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would ...
In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal I = m {\displaystyle I={\mathfrak {m}}} is especially important, for example the distinguished maximal ideal of a valuation ring .
Still more generally, if A is a regular local ring, then the formal power series ring A[[x]] is regular local. If Z is the ring of integers and X is an indeterminate, the ring Z[X] (2, X) (i.e. the ring Z[X] localized in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.
The idea is to approximate a surface part embedded into 3D non-plane contour by an arbitrary triangle grid. To converge such triangle grid to grid with minimum area one should solve the same two systems described above. Increments of the third nodal co-ordinates may be determined additionally by similar system at axis 3 in the following way
where is the maximum force predicted by the Lennard-Jones potential and is the maximum separation obtained by matching the areas under the Dugdale and Lennard-Jones curves (see adjacent figure). This means that the attractive force is constant for z 0 ≤ z ≤ z 0 + h 0 {\displaystyle z_{0}\leq z\leq z_{0}+h_{0}} .
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The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.