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Problem 7 (Recognizing Rings)

Which of the following are rings? Which rings have a unity? Which are commutative?

  1. For each $n$, the set $\mathbb{Z}_n$ together with modular addition and multiplication.
  2. The set $2\mathbb{Z}$ together with regular addition and multiplication.
  3. The set $\mathbb{Q}[x]$ of all polynomials in the variable $x$ with coefficients in $\mathbb{Q}$ together with polynomial addition and multiplication.
  4. Let $M_2(\mathbb{Z})$ bet the set of 2 by 2 matrices with entries in $\mathbb{Z}$, together with the usual properties of matrix addition and matrix multiplication.
  5. Consider the set $\mathbb{R}^3$, together with the two binary operations of vector addition and the cross product.
  6. The set of all real valued functions $f:\mathbb{R}\to\mathbb{R}$, together with function addition $(f+g)(x)=f(x)+g(x)$ and function multiplication $(fg)(x)=f(x)g(x)$.

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