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This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.
Some Related Sentences
is and up
`` Amen '', said the Reverend Doran, grabbing his rifle propped up against a tombstone, `` and now my brethren, it would seem that our presence is required elsewhere ''.
It became the sole `` subject '' of `` international law '' ( a term which, it is pertinent to remember, was coined by Bentham ), a body of legal principle which by and large was made up of what Western nations could do in the world arena.
It is the gait of the human who must run to live: arms dangling, legs barely swinging over the ground, head hung down and only occasionally swinging up to see the target, a loose motion that is just short of stumbling and yet is wonderfully graceful.
On Fridays, the day when many Persians relax with poetry, talk, and a samovar, people do not, it is true, stream into Chehel Sotun -- a pavilion and garden built by Shah Abbas 2, in the seventeenth century -- but they do retire into hundreds of pavilions throughout the city and up the river valley, which are smaller, more humble copies of the former.
Unruly hair goes straight up from his forehead, standing so high that the top falls gently over, as if to show that it really is hair and not bristle.
When I try to work out my reasons for feeling that this passage is of critical significance, I come up with the following ideas, which I shall express very briefly here and revert to in a later essay.
Professionally a lawyer, that is to say associated with dignity, reserve, discipline, with much that is essentially middle-class, he is compelled by an impossible love to exhibit himself dressed up, disguised -- that is, paradoxically, revealed -- as a child, and, worse, as a whore masquerading as a child.
Was it supposed, perchance, that A & M ( vocational training, that is ) was quite sufficient for the immigrant class which flooded that part of the New England world in the post-Civil War period, the immigrants having been brought in from Southern Europe, to work in the mills, to make up for the labor shortage caused by migration to the West??
Presumably a cocktail party is expected to fulfill the host's desire to get together a number of people who are inadequately acquainted and thereby arrange for bringing the level of acquaintance up to adequacy for future cooperative endeavors.
This is done for simplicity of commands and to bring the hidden redundancy up to where misunderstanding has almost zero possibility.
A further regulation is that commands always go down, unaccompanied by statements, and statements always go up, unaccompanied by commands.
It is of the utmost importance to the people of America and of the world how their governing President `` ends up '' during the four years of his term.
But a modern Oedipus who is doomed because he cannot oppose his own childhood is only pathetic, and for renouncing the mystery in favor of psychological truth he gives up the claim on our sympathies.
`` Dear Miss Steichen: It is a very good letter you send me -- softens the intensity of this guerilla warfare I am carrying on up here.
Community decision makers must make up their minds whether a claim is acceptable to the larger community in terms of prevailing expectations regarding members of nation states.
but the possibility of this effort is bound up with that development of historical thought which is the greatest achievement of our civilization in the last two centuries, and it is utterly impossible to people in whom this development has not taken place.
is and isomorphism
In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).
* Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.
In a category with exponentials, using the isomorphism ( in computer science, this is called currying ), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:
Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.
In fact, when A is a commutative unital C *- algebra, the Gelfand representation is then an isometric *- isomorphism between A and C ( Δ ( A )).
Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.
The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.
Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative " up to " an isomorphism.
In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.
is and only
Bryn Mawr Drive is only two or three miles from the Spartan, and it took me less than five minutes to get there.
Actually, only two men know what the formula is, Blake and '' -- He stopped and looked at Thor's body.
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
This is the only case in modern history of a people of Britannic origin submitting without continued struggle to what they view as foreign domination.
Recognizing that the Rule of Law is `` a dynamic concept which should be employed not only to safeguard the civil and political rights of the individual in a free society '', the Congress asserted that it also included the responsibility `` to establish social, economic, educational and cultural conditions under which his legitimate aspirations and dignity may be realized ''.
They are huge areas which have been swept by winds for so many centuries that there is no soil left, but only deep bare ridges fifty or sixty yards apart with ravines between them thirty or forty feet deep and the only thing that moves is a scuttling layer of sand.
Others are confined to vast reservations, and not only does the Australian government justifiably not wish them to be viewed as exhibits in a zoo, but on their reservations they are extremely fugitive, shunning camps, coming together only for corroborees at which their strange culture comes to its highest pitch -- which is very low indeed.
`` Now that Bruno Walter is virtually in retirement and my dear friend Dimitri Mitropoulos is no longer with us, I am probably the only one -- with the possible exception of Leonard Bernstein -- who has this special affinity for and champions the works of Bruckner and Mahler ''.
Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.
The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.
I think it is essential, however, to pinpoint here the difference between the two concepts of sovereignty that went to war in 1861 -- if only to see better how imperative is our need today to clarify completely our far worse confusion on this subject.
To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.