definition of ring of \(n \times n\) ring matrices
Topics
About: ring
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% matrices space.
Target Context
- The reader will have a definition of ring of \(n \times n\) ring matrices.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( \{M\}\): \(= \text{ the } R \text{ matrices space }\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\(*M_n (R)\): \(= \{M \in \{M\} \vert M \in \{\text{ the } n \times n \text{ matrices }\}\}\)
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Conditions:
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2: Note
Let us see that \(M_n (R)\) is indeed a ring.
\(M_n (R)\) is an Abelian group under addition: addition is associative, the \(0\) matrix with all the components \(0\) is the \(0\) element, and for each \(M \in M_n (R)\), \(- M\) with \((- M)^j_l = - M^j_l\) is the inverse of \(M\).
\(M_n (R)\) is associative under multiplication, because any multiplications of ring matrices are associative, by the proposition that for any ring, the multiplications of any matrices over the ring are associative.
\(I \in M_n (R)\) as the diagonal matrix with each diagonal element \(1\) is the identity element, because \((M I)^j_m = M^j_l I^l_m = M^j_m\) and \((I M)^j_m = I^j_l M^l_m = M^j_m\).
So, \(M_n (R)\) is a monoid.
Multiplication of \(M_n (R)\) is distributive with respect to addition, by the proposition that for any ring, the multiplications of any matrices over the ring are distributive with respect to addition.
So, \(M_n (R)\) is a ring.
