Zero object (algebra)

<h2 id="properties">Properties</h2>
<table><tbody><tr><td>2↕ </td><td>                                          [                                                            0                                                                              0                                                      ]                                {\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}}}  </td><td>=</td><td>                                          [                                                                                                                                                                                                ]                                {\displaystyle {\begin{bmatrix}\,\\\,\end{bmatrix}}}  </td><td>[ ]</td><td> ‹0</td></tr><tr><td></td><td>↔1</td><td></td><td>^0</td><td>↔1</td><td></td></tr><tr><td colspan="6">Element of the zero space, written as empty <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> (rightmost one), is multiplied by 2×0 <a href="/facts/Empty_matrix/qa8FI4ko">empty matrix</a> to obtain 2-dimensional zero vector (leftmost). Rules of <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> are respected.</td></tr></tbody></table>
<p>The zero ring, zero module and zero vector space are the <a href="/facts/Zero_object/oeckvXRj">zero objects</a> of, respectively, the <a href="/facts/Category_of_pseudo-rings/pIwY32bc">category of pseudo-rings</a>, the <a href="/facts/Category_of_modules/AnnF9JKf">category of modules</a> and the <a href="/facts/Category_of_vector_spaces/AnnF9JKf">category of vector spaces</a>. However, the zero ring is not a zero object in the <a href="/facts/Category_of_rings/pIwY32bc">category of rings</a>, since there is no <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> of the zero ring in any other ring.
</p><p>The zero object, by definition, must be a terminal object, which means that a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> <i>A</i> → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0.
</p><p>The zero object, also by definition, must be an initial object, which means that a morphism {0} → <i>A</i> must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ <i>A</i>, called the <a href="/facts/Zero_vector/PvH2qhxh">zero vector</a> in vector spaces. This map is a <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a>, and hence its image is isomorphic to {0}. For modules and vector spaces, this <a href="/facts/Subset/SJGERmbA">subset</a> {0} ⊂ <i>A</i> is the only empty-generated <a href="/facts/Submodule/jP0LIhFL">submodule</a> (or 0-dimensional <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a>) in each module (or vector space) A.
</p>
<h3>Unital structures</h3>
<p>The {0} object is a <a href="/facts/Terminal_object/oeckvXRj">terminal object</a> 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 <a href="/facts/Initial_object/oeckvXRj">initial object</a> (and hence, a <i>zero object</i> in the <a href="/facts/Category_theory/6zS3DXPa">category-theoretical</a> sense) depend on exact definition of the <a href="/facts/Multiplicative_identity/9nss3xHY">multiplicative identity</a> 1 in a specified structure.
</p><p>If the definition of 1 requires that 1 ≠ 0, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. If mathematicians sometimes talk about a <a href="/facts/Field_with_one_element/rwGle3sW">field with one element</a>, this abstract and somewhat mysterious mathematical object is not a field.
</p><p>In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the <a href="/facts/Category_of_rings/pIwY32bc">category of rings</a> Ring the ring of <a href="/facts/Integer/PUwwrawx">integers</a> Z is the initial object, not {0}.
</p><p>If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
</p>
<h2 id="notation">Notation</h2>
<p>Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Nildimensional_space/ywL3gK7q">Nildimensional space</a></li>
<li><a href="/facts/Triviality_(mathematics)/AqzUxokN">Triviality (mathematics)</a></li>
<li><a href="/facts/Examples_of_vector_spaces/tNTbVppn">Examples of vector spaces</a></li>
<li><a href="/facts/Field_with_one_element/rwGle3sW">Field with one element</a></li>
<li><a href="/facts/Empty_semigroup/UKudol1V">Empty semigroup</a></li>
<li><a href="/facts/Zero_element/PvH2qhxh">Zero element</a></li>
<li><a href="/facts/List_of_zero_terms/PvH2qhxh">List of zero terms</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>David Sharpe (1987). <i>Rings and factorization</i>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. p. <a href="https://books.google.com/books?id=Nmg4AAAAIAAJ&pg=PA10">10</a> : <i>trivial ring</i>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-33718-6.</li>
<li>Barile, Margherita. <a href="https://mathworld.wolfram.com/TrivialModule.html">"Trivial Module"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li>Barile, Margherita. <a href="https://mathworld.wolfram.com/ZeroModule.html">"Zero Module"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

Zero object (algebra) open-in-new

Zero object (algebra)