Question: "Compute the characteristic of the ring Z18×Z15"

High School
Geometry
High school geometry
Circles

Rivka Thorpe

Answered question

2021-02-01

Compute the characteristic of the ring $Z 18 \times Z 15$

Answer & Explanation

bahaistag

Skilled2021-02-02Added 100 answers

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

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