May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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Algebraic Structures 7-41
(1) (M, +) is an abelian group.
00
00
is the identity element and
−−
−−
ab
cd
is the inverse
of
ab
cd
∈ M.
(2) (M, ·) is a semigroup as matrix multiplication is associative.
(3) Matrix multiplication is distributive with respect to addition over a set of square matrices M.
Hence (M, +, ·) is a ring.
Since matrix multiplication is not commutative, it is not a commutative ring.
There is a unit element, which is identity matrix
10
01
.
Hence, it possesses unity element.
Zero Divisors: Let a ≠ 0, b ≠ 0 be two elements of a ring R. Such that a.b = 0, then a and b are
called proper divisors of zero.
Note 7.8 In a ring, it is possible to have ab = 0 when neither a nor b is zero.
In Example 4. of matrices( M, ...
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