Is the set of 2×2 matrices a ring?

The algebra M2(R) of 2 × 2 real matrices is a simple example of a non-commutative associative algebra. Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: E11E21 = 0, hence it is not a division ring.

What is a zero divisor of a ring?

A nonzero element of a ring for which , where is some other nonzero element and the multiplication is the multiplication of the ring. A ring with no zero divisors is known as an integral domain.

Can a ring have zero divisors?

Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no zero divisors except 0. A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.

How do you find the zeros divisor of a matrix?

You fill out v into a square matrix (with zeros even, if you like) and you’ve shown A is a zero divisor. It’s pretty trivial to see, then, that a matrix over A any ring R is a zero divisor iff there exists a nonzero vector v over R (with the right length) such that Av=0.

Are matrices ring?

For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. This idea can be used to represent operators on Hilbert spaces, for example.

How do you find the zero divisors of Z20?

Exercise 13-4 List all zero divisors of Z20: Observe that: 2 × 10 = 20 ≡ 0(mod 20) 4 × 5 = 20 ≡ 0(mod 20) 5 × 8 = 40 ≡ 0(mod 20) 6 × 10 = 60 ≡ 0(mod 20) 8 × 5 = 40 ≡ 0(mod 20) 10 × 8 = 80 ≡ 0(mod 20) 12 × 10 = 120 ≡ 0(mod 20) 14 × 10 = 140 ≡ 0(mod 20) 15 × 4 = 60 ≡ 0(mod 20) 16 × 5 = 80 ≡ 0(mod 20) 18 × 10 = 180 ≡ 0( …

What are the zero divisors of Z12?

(ii) There are 7 zero divisors in Z12 : namely, 2, 3, 4, 6, 8, 9, 10, since 2 · 6=0, 3 · 4=0, 3 · 8=0, 4 · 9=0, 6 · 10 = 0.

What are the zero divisors of Z20?

The zero divisors in Z20 are {2,4, 5,6,8, 10,12,14,15,16,18}. Every nonzero element is either a zero divisor or a unit.

Do diagonal matrices form a ring?

The sum and product of diagonal matrices are again diagonal, and the diagonal matrices form a subring of the ring of square matrices: indeed for n×n matrices over a ring R this ring is isomorphic to the product ring Rn.