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50 points. Can anyone give an example of a noncommutative ring that doesn't involve matrices?

User Jimw
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Hamilton's quaternions are an example of non-commutative ring. If you are not familiar with them, they are similar to complex numbers, except for the fact that there are three roots of -1: the ring is generated by


H = <1, i, j, k>

Which means that a generic element is appears as follows:


z = a+bi+cj+dk,\quad a,b,c,d \in \mathbb{R}

The rules for additions are the usual: you sum like terms. As for the mutliplicaitons, you work like this:


ij = k,\quad kj = i,\quad ki = j,\quad ji = -k,\quad kj = -i,\quad ik = -j

User Jojo Joseph
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