# Compute the characteristic of the ring Z18 × Z15

Question
Circles
Compute the characteristic of the ring Z18 × Z15

2021-02-02
Number of element in $$\displaystyle{Z}{18}×{Z}{15}$$ is $$\displaystyle{18}×{15}={270}.$$
So, 270x will be the zero element of $$\displaystyle{Z}{18}×{Z}{15}={\left(\begin{array}{cc} {0}&{0}\end{array}\right)}{Z}={\left(\begin{array}{cc} {0}&{0}\end{array}\right)}$$ of for all $$\displaystyle{x}\in{Z}{18}×{Z}{15}.$$ Let nn is charactiristic of Z18×Z15 then nn must divide 270 and again let $$\displaystyle{\left({1},{1}\right)}\in{Z}{18}×{Z}{15}{\left({1},{1}\right)}\in{Z}$$ then $$\displaystyle{n}{\left({1},{1}\right)}={\left({n},{n}\right)}≡{\left({0},{0}\right)}\in{Z}{18}×{Z}{15}{Z}$$, that is nn must be divisible by 18 and 15, infact nn must be least common multiple of 15 and 18. So, charactiristic of $$\displaystyle{Z}{18}×{Z}{15}={l}.{c}.{m}{\left({18},{15}\right)}={90}.$$

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