View source for Matrix ring

{{Short description|Mathematical ring whose elements are matrices}}
{{Redirect|Matrix algebra|the algebraic theory of matrices|Matrix (mathematics)|and|Linear algebra}}

In [[abstract algebra]], a '''matrix ring''' is a set of [[matrix (mathematics)|matrices]] with entries in a [[ring (mathematics)|ring]] ''R'' that form a ring under [[matrix addition]] and [[matrix multiplication]].{{sfnp|Lam|1999|loc=Theorem 3.1|ps=}} The set of all {{nowrap|''n'' × ''n''}} matrices with entries in ''R'' is a matrix ring denoted M<sub>''n''</sub>(''R''){{sfnp|Lam|2001|}}{{sfnp|Lang|2005|loc=V.§3|ps=}}{{sfnp|Serre|2006|p=3|ps=}}{{sfnp|Serre|1979|p=158|ps=}} (alternative notations: Mat<sub>''n''</sub>(''R''){{sfnp|Lang|2005|loc=V.§3|ps=}} and {{nowrap|''R''<sup>''n''×''n''</sup>}}{{sfnp|Artin|2018|loc=Example 3.3.6(a)|ps=}}).  Some sets of infinite matrices form '''infinite matrix rings'''.  A subring of a matrix ring is again a matrix ring.  Over a [[rng (algebra)|rng]], one can form matrix rngs.

When ''R'' is a commutative ring, the matrix ring M<sub>''n''</sub>(''R'') is an [[associative algebra]] over ''R'', and may be called a '''matrix algebra'''.  In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''.

== Examples ==
* The set of all {{nowrap|''n'' × ''n''}} [[square matrices]] over ''R'', denoted M<sub>''n''</sub>(''R'').  This is sometimes called the "full ring of ''n''-by-''n'' matrices".
* The set of all upper [[triangular matrices]] over ''R''.
* The set of all lower [[triangular matrices]] over ''R''.
* The set of all [[diagonal matrices]] over ''R''.  This [[subalgebra]] of M<sub>''n''</sub>(''R'') is [[algebra homomorphism|isomorphic]] to the [[product of rings|direct product]] of ''n'' copies of ''R''.
* For any index set ''I'', the ring of endomorphisms of the right ''R''-module <math display="inline">M=\bigoplus_{i\in I}R</math> is isomorphic to the ring <math>\mathbb{CFM}_I(R)</math>{{fact|date=December 2020}}<!--Reference for this notation?--> of '''column finite matrices''' whose entries are indexed by {{nowrap|''I'' × ''I''}} and whose columns each contain only finitely many nonzero entries.  The ring of endomorphisms of ''M'' considered as a left ''R''-module is isomorphic to the ring <math>\mathbb{RFM}_I(R)</math> of '''row finite matrices'''.
* If ''R'' is a [[Banach algebra]], then the condition of row or column finiteness in the previous point can be relaxed.  With the norm in place, [[absolutely convergent series]] can be used instead of finite sums.  For example, the matrices whose column sums are absolutely convergent sequences form a ring.{{dubious|date=December 2020}}<!--Seems wrong, without any control on the growth rate in a row.-->  Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.{{dubious|date=December 2020}}  This idea can be used to represent [[Hilbert space#Operators on Hilbert spaces|operators on Hilbert spaces]], for example.
* The intersection of the row-finite and column-finite matrix rings forms a ring <math>\mathbb{RCFM}_I(R)</math>.
* If ''R'' is [[commutative ring|commutative]], then M<sub>''n''</sub>(''R'') has a structure of a [[*-algebra]] over ''R'', where the [[involution (mathematics)#Ring theory|involution]] * on M<sub>''n''</sub>(''R'') is [[matrix transpose|matrix transposition]]. 
* If ''A'' is a [[C*-algebra]], then M<sub>''n''</sub>(''A'') is another C*-algebra. If ''A'' is non-unital, then M<sub>''n''</sub>(''A'') is also non-unital. By the [[Gelfand–Naimark theorem]], there exists a [[Hilbert space]] ''H'' and an isometric *-isomorphism from ''A'' to a norm-closed subalgebra of the algebra ''B''(''H'') of continuous operators; this identifies M<sub>''n''</sub>(''A'') with a subalgebra of ''B''(''H''<sup>⊕''n''</sup>). For simplicity, if we further suppose that ''H'' is separable and ''A'' <math>\subseteq</math> ''B''(''H'') is a unital C*-algebra, we can break up ''A'' into a matrix ring over a smaller C*-algebra. One can do so by fixing a [[Projection (linear_algebra)#Orthogonal projections|projection]] ''p'' and hence its orthogonal projection 1&nbsp;−&nbsp;''p''; one can identify ''A'' with <math display="inline">\begin{pmatrix} pAp & pA(1-p) \\ (1-p)Ap & (1-p)A(1-p) \end{pmatrix}</math>, where matrix multiplication works as intended because of the orthogonality of the projections. In order to identify ''A'' with a matrix ring over a C*-algebra, we require that ''p'' and 1&nbsp;−&nbsp;''p'' have the same "rank"; more precisely, we need that ''p'' and 1&nbsp;−&nbsp;''p'' are Murray–von&nbsp;Neumann equivalent, i.e., there exists a [[partial isometry]] ''u'' such that {{nowrap|1=''p'' = ''uu''*}} and {{nowrap|1=1 − ''p'' = ''u''*''u''}}. One can easily generalize this to matrices of larger sizes.
* Complex matrix algebras M<sub>''n''</sub>('''C''') are, up to isomorphism, the only finite-dimensional simple associative algebras over the field '''C''' of [[complex number]]s. Prior to the invention of matrix algebras, [[William Rowan Hamilton|Hamilton]] in 1853 introduced a ring, whose elements he called [[biquaternions]]<ref>Lecture VII of Sir William Rowan Hamilton (1853) ''Lectures on Quaternions'', Hodges and Smith</ref> and modern authors would call tensors in {{nowrap|'''C''' ⊗<sub>'''R'''</sub> '''H'''}}, that was later shown to be isomorphic to M<sub>2</sub>('''C''').  One [[basis (linear algebra)|basis]] of M<sub>2</sub>('''C''') consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by the [[identity matrix]] and the three [[Pauli matrices]].
* A matrix ring over a field is a [[Frobenius algebra]], with Frobenius form given by the trace of the product: {{nowrap|1=''σ''(''A'', ''B'') = tr(''AB'')}}.

== Structure ==
* The matrix ring M<sub>''n''</sub>(''R'') can be identified with the [[ring of endomorphisms]] of the [[free module|free right ''R''-module]] of rank ''n''; that is, {{nowrap|M<sub>''n''</sub>(''R'') ≅ End<sub>''R''</sub>(''R''<sup>''n''</sup>)}}.  [[Matrix multiplication]] corresponds to composition of endomorphisms.
* The ring M<sub>''n''</sub>(''D'') over a [[division ring]] ''D'' is an [[Artinian ring|Artinian]] [[simple ring]], a special type of [[semisimple ring]].  The rings <math>\mathbb{CFM}_I(D)</math> and <math>\mathbb{RFM}_I(D)</math> are ''not'' simple and not Artinian if the set ''I'' is infinite, but they are still [[full linear ring]]s.
* The [[Artin–Wedderburn theorem]] states that every semisimple ring is isomorphic to a finite [[direct product]] <math display="inline">\prod_{i=1}^r \operatorname{M}_{n_i}(D_i)</math>, for some nonnegative integer ''r'', positive integers ''n''<sub>''i''</sub>, and division rings ''D''<sub>''i''</sub>.
* When we view M<sub>''n''</sub>('''C''') as the ring of linear endomorphisms of '''C'''<sup>''n''</sup>, those matrices which vanish on a given subspace ''V'' form a [[left ideal]]. Conversely, for a given left ideal ''I'' of M<sub>''n''</sub>('''C''') the intersection of [[Kernel (linear algebra)|null spaces]] of all matrices in ''I'' gives a subspace of '''C'''<sup>''n''</sup>. Under this construction, the left ideals of M<sub>''n''</sub>('''C''') are in bijection with the subspaces of '''C'''<sup>''n''</sup>.
* There is a bijection between the two-sided [[ideal (ring theory)|ideals]] of M<sub>''n''</sub>(''R'') and the two-sided ideals of ''R''.  Namely, for each ideal ''I'' of ''R'', the set of all {{nowrap|''n'' × ''n''}} matrices with entries in ''I'' is an ideal of M<sub>''n''</sub>(''R''), and each ideal of M<sub>''n''</sub>(''R'') arises in this way.  This implies that M<sub>''n''</sub>(''R'') is [[simple ring|simple]] if and only if ''R'' is simple. For {{nowrap|''n'' ≥ 2}}, not every left ideal or right ideal of M<sub>''n''</sub>(''R'') arises by the previous construction from a left ideal or a right ideal in ''R''. For example, the set of matrices whose columns with indices 2 through ''n'' are all zero forms a left ideal in M<sub>''n''</sub>(''R'').
* The previous ideal correspondence actually arises from the fact that the rings ''R'' and M<sub>''n''</sub>(''R'') are [[Morita equivalent]]. Roughly speaking, this means that the category of left ''R''-modules and the category of left M<sub>''n''</sub>(''R'')-modules are very similar. Because of this, there is a natural bijective correspondence between the ''isomorphism classes'' of left ''R''-modules and left M<sub>''n''</sub>(''R'')-modules, and between the isomorphism classes of left ideals of ''R'' and left ideals of M<sub>''n''</sub>(''R''). Identical statements hold for right modules and right ideals.  Through Morita equivalence, M<sub>''n''</sub>(''R'') inherits any [[Morita equivalence|Morita-invariant]] properties of ''R'', such as being [[simple ring|simple]], [[Artinian ring|Artinian]], [[Noetherian ring|Noetherian]], [[Prime ring|prime]].

== Properties ==
* If ''S'' is a [[subring]] of ''R'', then M<sub>''n''</sub>(''S'') is a subring of M<sub>''n''</sub>(''R''). For example, M<sub>''n''</sub>('''Z''') is a subring of M<sub>''n''</sub>('''Q''').
* The matrix ring M<sub>''n''</sub>(''R'') is [[commutative ring|commutative]] if and only if {{nowrap|1=''n'' = 0}}, {{nowrap|1=''R'' = 0}}, or ''R'' is [[commutative ring|commutative]] and {{nowrap|1=''n'' = 1}}. In fact, this is true also for the subring of upper triangular matrices.  Here is an example showing two upper triangular {{nowrap|2 × 2}} matrices that do not commute, assuming {{nowrap| 1 ≠ 0}} in ''R'':
*:: <math>
  \begin{bmatrix}
    1 & 0 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 1 \\
    0 & 0
  \end{bmatrix}
</math>
*: and
*:: <math>  \begin{bmatrix}
    1 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix}.
</math>
* For {{nowrap|''n'' ≥ 2}}, the matrix ring M<sub>''n''</sub>(''R'') over a [[zero ring|nonzero ring]] has [[zero divisor]]s and [[nilpotent element]]s; the same holds for the ring of upper triangular matrices. An example in {{nowrap|2 × 2}} matrices would be
*:: <math>  \begin{bmatrix}
    0 & 1 \\
    0 & 0 
  \end{bmatrix}
  \begin{bmatrix}
    0 & 1 \\
    0 & 0
  \end{bmatrix}
=
  \begin{bmatrix}
    0 & 0 \\
    0 & 0
  \end{bmatrix}.
</math>
* The [[center (ring theory)|center]] of M<sub>''n''</sub>(''R'') consists of the scalar multiples of the [[identity matrix]], ''I''<sub>''n''</sub>, in which the scalar belongs to the center of ''R''.
* The [[unit group]] of M<sub>''n''</sub>(''R''), consisting of the invertible matrices under multiplication, is denoted GL<sub>''n''</sub>(''R'').
* If ''F'' is a field, then for any two matrices ''A'' and ''B'' in M<sub>''n''</sub>(''F''), the equality {{nowrap|1=''AB'' = ''I''<sub>''n''</sub>}} implies {{nowrap|1=''BA'' = ''I''<sub>''n''</sub>}}.  This is not true for every ring ''R'' though.  A ring ''R'' whose matrix rings all have the mentioned property is known as a [[stably finite ring]] {{harv|Lam|1999|p=5}}.

== Matrix semiring ==
In fact, ''R'' needs to be only a [[semiring]] for M<sub>''n''</sub>(''R'') to be defined. In this case, M<sub>''n''</sub>(''R'') is a semiring, called the '''matrix semiring'''. Similarly, if ''R'' is a commutative semiring, then M<sub>''n''</sub>(''R'') is a '''{{visible anchor|matrix semialgebra}}'''.

For example, if ''R'' is the [[Boolean semiring]] (the [[two-element Boolean algebra]] {{nowrap|1=''R'' = {{mset|0, 1}}}} with {{nowrap|1=1 + 1 = 1}}),{{sfnp|Droste|Kuich|2009|p=7|ps=}} then M<sub>''n''</sub>(''R'') is the semiring of [[binary relation]]s on an ''n''-element set with union as addition, [[composition of relations]] as multiplication, the [[empty relation]] ([[zero matrix]]) as the zero, and the [[identity relation]] ([[identity matrix]]) as the [[identity element|unity]].{{sfnp|Droste|Kuich|2009|p=8|ps=}}

== See also ==
* [[Central simple algebra]]
* [[Clifford algebra]]
* [[Hurwitz's theorem (normed division algebras)]]
* [[Generic matrix ring]]
* [[Sylvester's law of inertia]]

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation | last1=Artin | year=2018 | title=Algebra | publisher=Pearson |author-link=Michael Artin}}
* {{citation | last1=Droste | first1=M. | last2=Kuich | first2=W | year=2009 | chapter=Semirings and Formal Power Series | title=Handbook of Weighted Automata | series=Monographs in Theoretical Computer Science. An EATCS Series | pages=3–28 | doi=10.1007/978-3-642-01492-5_1 | isbn=978-3-642-01491-8 }}
* {{citation | last1=Lam | first1=T. Y. | year=1999 | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | url=https://books.google.com/books?id=r9VoYbk-8c4C&q=%22matrix+ring%22 |author-link=Tsit Yuen Lam}}
* {{citation | last1=Lam | year=2001 | title=A first course on noncommutative rings | edition=2nd | publisher=Springer }}
* {{citation | last1=Lang | year=2005 | title=Undergraduate algebra | publisher=Springer |author-link=Serge Lang}}
* {{citation | last1=Serre | year=1979 | title=Local fields | publisher=Springer |author-link=Jean-Pierre Serre }}
* {{citation | last1=Serre | year=2006 | title=Lie algebras and Lie groups | edition=2nd | publisher=Springer }}, corrected 5th printing
{{refend}}

[[Category:Algebraic structures]]
[[Category:Ring theory]]
[[Category:Matrix theory]]