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Abstract Algebra

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Moderate

Abstract Algebra: What is an Algebra in the Abstract Setting?

ABSALG-UWFP9H

Let $R$ be a commutative ring with identity $1_R$.

An $R$-algebra $\mathcal{A}$ is a ring $(\mathcal{A}, +, \cdot)$ which is an $R$-module, such that $r(a\cdot a')=(ra)\cdot a'=a\cdot (ra')$ and $1_Ra=a$ , for all $r\in R$ and $a,a'\in \mathcal{A}$. Here $(r,a)\mapsto ra$ is the $R$-module action, and $+$, $\cdot$ are, respectively, the addition and multiplication in the ring $\mathcal{A}$.

Which of the following are true?

Select ALL that apply.

A

$(\mathcal{A}, +)$ is always an abelian group.

B

$(\mathcal{A}, \cdot)$ is always a group.

C

$(\mathcal{A}, + ,\cdot)$ is always a commutative ring.

D

The $n\times n$ matrices with entries in $\mathbb{R}$ form an $\mathbb{R}$-algebra.​

E

The $\mathbb{R}$-vector space $\mathbb{R}^3$, with $\cdot=$Cross Product, is an $\mathbb{R}$-algebra.

F

The $\mathbb{R}$-vector space $\mathbb{R}^3$, with $\cdot=$Dot Product, is an $\mathbb{R}$-algebra.

G

The ​complex numbers $\mathbb{C}$ form an $\mathbb{R}$-algebra.

H

A ring $R$ is itself a $\mathbb{Z}$-algebra.