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Page "Endomorphism ring" ¶ 10
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If and module
If a particular object did not support a, it could be easily added in the module.
If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.
If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.
If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism so that, or f can also be viewed as a left R module homomorphism, where.
If M is a free module over a principal ideal domain R, then every submodule of M is again free.
If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.
If k is a field and G is a group, then a group representation of G is a left module over the group ring k. The simple k modules are also known as irreducible representations.
If M is a module which has a non-zero proper submodule N, then there is a short exact sequence
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 Jacobson radical J ( R ).
* If L is a maximal left ideal, then R / L is a simple left R module.
If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.
The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod ( C ) := Add ( C, Ab ) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of ( left ) R-modules.
If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.
If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
* If an R module is finitely generated and projective ( that is, a progenerator ), then the endomorphism ring of the module and R share all Morita invariant properties.
If we interpret the object as the left module, then this matrix category becomes a subcategory of the category of left modules over.
If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A ; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
If H is a left module over the ring R, one forms the ( algebraic ) character module H * consisting of all abelian group homomorphisms from H to Q / Z.
If F ( U ) is a module over the ring O < sub > X </ sub >( U ) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an O < sub > X </ sub >- module.

If and is
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.
If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;
If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If it is an honest feeling, then why should she not yield to it??
If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If he is good, he may not be legal ; ;
If the man on the sidewalk is surprised at this question, it has served as an exclamation.
If the existent form is to be retained new factors that reinforce it must be introduced into the situation.
If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??
If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.
If he is a traditionalist, he is an eclectic traditionalist.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If and simple
If **ya is the multiple secant of **zg which passes through Af and **yb is the simple secant of **zg which passes through Af, and if Af are the points in which **ya meets **zg, and if Af is the image of Af on the generator **yb, it follows that the image of the line Af is Af.
If, however, the figure to be discerned were complicated, composed of several interlocking subfigures, and so on, even the tracing process failed him, and he could not focus even relatively simple shapes among its parts.
If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.
If he foresaw any problem because of the quality of the hymen, it was recommended that simple procedures be undertaken at once to incise the hymen or, preferably, to dilate it.
If we have not thought and made a decision entirely in these terms, then we need to submit ourselves to the following `` simple test '': `` Have we decided how we are to kill the other members of our household in the event of our being less injured than they are ''??
If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.
If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.
If sections of the frame move in a simple manner, the compressor emits a ( slightly longer ) command that tells the decompresser to shift, rotate, lighten, or darken the copy: a longer command, but still much shorter than intraframe compression.
If is a constant, the solution is particularly simple, and describes, e. g., if, the exponential decay of radioactive material at the macroscopic level.
If the deficit is not financed through a loan it is enough with a simple majority to approve the budget ( 50 % of the votes plus 1 ).
If the outcomes x < sub > i </ sub > are not equiprobable, then the simple average ought to be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others.
If it is unaffected, then it is a simple partial seizure ; otherwise it is a complex partial ( psychomotor ) seizure.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency ( which does not depend on the amplitude ).
If it were simple to ascertain the properties required to develop a lead discovered in vitro to one that is active in vivo, drug discovery would be as reliable as drug manufacturing.
If the true function is simple, then an " inflexible " learning algorithm with high bias and low variance will be able to learn it from a small amount of data.
A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.
If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.
If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon.
If chitin is detected, they then produce enzymes to digest it by cleaving the glycosidic bonds in order to convert it to simple sugars and ammonia.
If they both used the Attic stadion of about 185 meters, then the older estimate is 1 / 6 too large, and Ptolemy's value is 1 / 6 too small, a difference explained as due to ancient scientists ' use of simple methods of measuring the earth, which were corrupted either high or low by a factor of 5 / 6, due to air's bending of horizontal light rays by 1 / 6 of the Earth's curvature.
If adding that detail only altered things slightly then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to infinite probability amplitudes.
If W isn't irreducible, then the dominant species ( or quasispecies ) that develops can depend on the initial population, as is the case in the simple example given below.
The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.
If they had any significant level of non-organic parts then something as simple as an x-ray scan would show them not to be human ( unless their creators designed the non-organic parts to have almost exactly the same structure and density as the organic components they were replacing, which would be both infeasible and highly improbable but not completely out of the question ).

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