Tagged: field

The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$

Problem 283

Let $F$ be a field and let
\[H(F)=\left\{\, \begin{bmatrix}
1 & a & b \\
0 &1 &c \\
0 & 0 & 1
\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

 
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The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain

Problem 198

Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.

Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.

 
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