The primary decomposition theorem

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

( primary decomposition theorem ).

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

In concluding this section, we should like to give an example which illustrates some of the ideas of the primary decomposition theorem.

This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

Research in community ecology might measure primary production in a wetland in relation to decomposition and consumption rates.

This is referred to as a primary decomposition of the ideal ( n ).

Thus, a primary decomposition of ( n ) corresponds to representing ( n ) as the intersection of finitely many primary ideals.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor − 1 also asserts uniqueness of the representation for p < sub > i </ sub > prime and e < sub > i </ sub > positive, a primary decomposition of ( n ) is essentially unique.

For any primary decomposition of I, the set of all radicals, that is, the set remains the same by the Lasker – Noether theorem.

The nitrogen cycle is of particular interest to ecologists because nitrogen availability can affect the rate of key ecosystem processes, including primary production and decomposition.

* Primary decomposition, a decomposition into primary ideals

In most grassland ecosystems, natural damage from fire, insects that feed on decaying matter, termites, grazing mammals, and the physical movement of animals through the grass are the primary agents of breakdown and nutrient cycling, while bacteria and fungi play the main roles in further decomposition.

Lasker was also a research mathematician who was known for his contributions to commutative algebra, which included proving the primary decomposition of the ideals of polynomial rings.

There is equilibrium between gross primary production and total respiration, between energy used from sunlight and energy released by decomposition, between uptake of nutrients from the soil and the return of nutrient by litter fall to the soil.

One of the primary roles of fungi in the ecosystem is decomposition, which is performed by the mycelium.

The primary cause of the delivery was the otherwise normal contractions, which had begun prior to death, and was therefore not related to processes of decomposition.

the primary decomposition is a decomposition into indecomposable modules,

Natural uranium is also a primary energy source extracted from the earth but does not come from the decomposition of organisms ( mineral fuel ).

It is based on the fact that the vector space can be canonically decomposed into a direct sum of stable subspaces corresponding to the distinct irreducible factors P of the characteristic polynomial ( as stated by the lemme des noyaux ), where the characteristic polynomial of each summand is a power of the corresponding P. These summands can be further decomposed, non-canonically, as a direct sum of cyclic F-modules ( like is done for the Frobenius normal form above ), where the characteristic polynomial of each summand is still a ( generally smaller ) power of P. The primary rational canonical form is a block diagonal matrix corresponding to such a decomposition into cyclic modules, with a particular form called generalized Jordan block in the diagonal blocks, corresponding to a particular choice of a basis for the cyclic modules.

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker – Noether theorem.

It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen.

The equivalence theorem also is a springboard for Coase's primary achievement — providing the pillars for the New Institutional Economics.

In mathematics, the Lasker – Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection of finitely many primary ideals ( which are related to, but not quite the same as, powers of prime ideals ).

This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of ( irreducible ) varieties.

The Lasker – Noether theorem for modules states every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.

Pattern-directed invocation is a commonly used and more practical alternative to first-order theorem proving in automated theorem proving, and is the primary method of implementing a blackboard system for automated reasoning in general.

Too many people think that the primary purpose of a higher education is to help you make a living ; ;

This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

The principal defender of this view of primary experience as `` causal efficacy '' is Alfred North Whitehead.

Speaking as a non-Jew I believe that its primary contribution is in the realm of future policy.

The natural and primary aesthetic attitude is to enjoy contemporary art, to despise and dislike the art of the recent past, and wholly to ignore everything else ''.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

But competent observers believe he is making progress, particularly toward what Sen. Jackson lists as the primary need -- `` a clearer understanding of where our vital national interests lie and what we must do to promote them ''.

Its primary advantage is that it is a moral choice ; ;

A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.

If the Department of State is to take primary responsibility for foreign policy in Washington, it follows that the ambassador is expected to take charge overseas.

For this reason, the more uncertain skywave service was denominated `` secondary '' in our rules, as compared to the steadier, more reliable groundwave `` primary service '', and, for both skywave service and skywave interference, signal strength is expressed in terms of percentage of time a particular signal-intensity level is exceeded -- 50 percent of the time for skywave service, 10 percent of the time for skywave interference.

As mentioned, the primary allocation objective to be followed in the allocation of stations on clear channels is the provision of widespread service, free from destructive interference.

The result is a color picture of the specimen where the primary colors correspond to the three different ultraviolet wavelengths.

A primary ingredient in these fields is imagination, and Tri-State Pipeline Corporation deserves a very good mark.

Though there is obviously great need for continued experimentation with various types of short-term intervention to further efforts in developing an operational definition of prevention at the secondary -- or perhaps, in some instances, primary -- level, the place of short-term intervention has already been documented by a number of investigators in a wide variety of settings.