Compute the characteristic of the ring Z18 × Z15

Compute the characteristic of the ring $Z18×Z15$

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bahaistag

Number of element in $Z18×Z15$ is $18×15=270.$
So, 270x will be the zero element of $Z18×Z15=\left(\begin{array}{cc}0& 0\end{array}\right)Z=\left(\begin{array}{cc}0& 0\end{array}\right)$ of for all $x\in Z18×Z15.$ Let nn is charactiristic of $Z18×Z15$ then nn must divide 270 and again let $\left(1,1\right)\in Z18×Z15\left(1,1\right)\in Z$ then $n\left(1,1\right)=\left(n,n\right)\equiv \left(0,0\right)\in Z18×Z15Z$, that is nn must be divisible by 18 and 15, infact nn must be least common multiple of 15 and 18. So, charactiristic of $Z18×Z15=l.c.m\left(18,15\right)=90.$