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of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an
100:
72:
53:
902:, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
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These objects are described jointly not only based on the common singleton and trivial group structure, but also because of
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86:
17:
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If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor
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In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the
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132:
This article is about trivial or zero algebraic structures. For zero elements in algebraic structures, see
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object cannot exist because it may contain only one element. In particular, the zero ring is not a
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The zero object, also by definition, must be an initial object, which means that a morphism
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For structures requiring the multiplication structure inside the zero object, such as the
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object can exist. But not as initial object because identity-preserving morphisms from
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463:, since the single element is its own multiplicative inverse. Some properties of
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which has both an additive and multiplicative identity is trivial if and only if
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The zero object, by definition, must be a terminal object, which means that a
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is, in the sense explained below, the simplest object of such structure. As a
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1045:
675:
525:
380:
227:
183:
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861:, this abstract and somewhat mysterious mathematical object is not a field.
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Instances of the zero object include, but are not limited to the following:
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133:
772:
693:
384:
207:
186:. The aforementioned abelian group structure is usually identified as
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depend on exact definition of the multiplicative identity; see
379:, because there are no non-zero elements. This structure is
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Zero vector spaces and zero modules are usually denoted by
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The zero ring, zero module and zero vector space are the
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to obtain 2-dimensional zero vector (leftmost). Rules of
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by an element of the base ring (or field) is defined as:
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is an especially ubiquitous example of a zero object, a
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must exist and be unique for an arbitrary object
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must exist and be unique for an arbitrary object
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Any trivial algebra is also a trivial ring. A trivial
708:. However, the zero ring is not a zero object in the
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592:{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}}}
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918:). This is always the case when they occur in an
480:is simultaneously a zero vector space considered
286:is also used, although it may be ambiguous, as a
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360:The most general of them, the zero module, is a
194:, so the object itself is typically denoted as
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779:, and hence its image is isomorphic to
674:Element of the zero space, written as empty
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857:. If mathematicians sometimes talk about a
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395:, since this equality implies that for all
210:to any other (under a unique isomorphism).
834:sense) depend on exact definition of the
734:. This morphism maps any element of
623:
618:
136:. For the zero object in a category, see
120:Learn how and when to remove this message
27:Algebraic structure with only one element
802:) in each module (or vector space)
449:{\displaystyle r=r\times 1=r\times 0=0.}
142:
499:. A trivial algebra is an example of a
14:
1044:
783:. For modules and vector spaces, this
678:(rightmost one), is multiplied by 2Ă0
469:
459:In this case it is possible to define
327:shared category-theoretical properties
1025:
1005:
809:
716:of the zero ring in any other ring.
495:The trivial ring is an example of a
58:adding citations to reliable sources
29:
24:
876:do not exist. For example, in the
838: 1 in a specified structure.
25:
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492:is simultaneously a zero module.
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190:, and the only element is called
891:is the initial object, not
775:in vector spaces. This map is a
206:) since every trivial object is
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202:trivial object (of a specified
45:needs additional citations for
375:, there is only one possible,
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1:
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315:zero-dimensional vector space
794:is the only empty-generated
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332:In the last three cases the
182:structure, which is also an
138:Initial and terminal objects
7:
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905:
764:, to the zero element
760:, the only element of
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516:over a field with an empty
150:to and from the zero object
10:
1083:
980:Cambridge University Press
841:If the definition of
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69:"Zero object" algebra
1067:Objects (category theory)
942:Examples of vector spaces
706:category of vector spaces
673:
362:finitely-generated module
937:Triviality (mathematics)
698:category of pseudo-rings
470:§ Unital structures
976:Rings and factorization
836:multiplicative identity
947:Field with one element
859:field with one element
696:of, respectively, the
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294:with a trivial action.
198:. One often refers to
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974:David Sharpe (1987).
756:. This morphism maps
684:matrix multiplication
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506:The zero-dimensional
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334:scalar multiplication
146:
1026:Barile, Margherita.
932:Nildimensional space
872:to any object where
832:category-theoretical
712:, since there is no
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560:
478:algebra over a field
410:
253:algebra over a field
54:improve this article
702:category of modules
524:zero. It is also a
520:. It therefore has
257:algebra over a ring
164:algebraic structure
1007:Barile, Margherita
962:List of zero terms
798:(or 0-dimensional
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497:rng of square zero
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878:category of rings
810:Unital structures
714:ring homomorphism
710:category of rings
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16:(Redirected from
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110:February 2012
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65:Find sources:
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49:
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43:This article
41:
37:
32:
31:
19:
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1014:
988:trivial ring
987:
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957:Zero element
914:(instead of
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883:the ring of
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818:object is a
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777:monomorphism
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694:zero objects
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680:empty matrix
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514:vector space
509:vector space
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501:zero algebra
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488:, a trivial
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134:Zero element
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52:Please help
47:verification
44:
1052:Ring theory
849:, then the
828:zero object
773:zero vector
385:commutative
381:associative
282:. The term
280:zero module
174:, and as a
162:of a given
160:zero object
1062:0 (number)
1046:Categories
982:. p.
543:Properties
319:zero space
223:zero group
208:isomorphic
80:newspapers
1033:MathWorld
1016:MathWorld
796:submodule
646: âč0
522:dimension
484:. Over a
435:×
423:×
387:. A ring
377:0 Ă 0 = 0
241:zero ring
172:singleton
148:Morphisms
986: :
926:See also
906:Notation
885:integers
738:to
721:morphism
704:and the
532:, and a
530:addition
364:with an
346:, where
317:or just
301:(over a
292:G-module
270:(over a
204:category
188:addition
170:it is a
830:in the
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490:algebra
473:below.
399:within
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180:trivial
156:algebra
94:scholar
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847:1 â 0
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482:below
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366:empty
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297:As a
290:is a
266:As a
234:As a
219:group
217:As a
176:magma
101:JSTOR
87:books
992:ISBN
881:Ring
814:The
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383:and
272:ring
236:ring
192:zero
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