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In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness?

I know $\mathbb{Z}$, $\mathbb{Z}[X]$, $K[X]$ and $\mathbb{Z}[\sqrt{d}], d \in \mathbb{Z}$ which are all quite abstract. Maybe not $\mathbb{Z}$ but it is too "nice" and doesn't illustrate the more interesting properties of rings or the lack thereof very well.

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  • $\begingroup$ You also have $\mathbb{Z}/n\mathbb{Z}$ but maybe it's a little abstract too. $\endgroup$ – Augustin May 26 '15 at 20:45
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    $\begingroup$ Many rings can be represente by matrices, and $2\times 2$ matrices represents linear tranformations in $\mathbb{R}^2$ that can be easely visualized and have many interesting properties of rings. $\endgroup$ – Emilio Novati May 26 '15 at 20:48
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    $\begingroup$ The ring of $n \times n$ matrices over a field under addition and matrix multiplication encodes transformations of a vector space over that field. Endomorphisms (homomorphisms from a group to itself) of abelian groups form a ring under pointwise addition and composition; they encode symmetries of a group. $\endgroup$ – Zach Effman May 26 '15 at 20:49
  • $\begingroup$ finite dimensional algebras like the complex numbers and dual numbers are good examples of rings of the form $\mathbb R[X]/f(X)\mathbb R[X]$. $\endgroup$ – jkabrg May 26 '15 at 20:49
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If you want an easy to understand new concrete example of associative rules, distributive rule, and multiplication inverses and identity rules that contribute to defining a ring, consider the so-called "tropical semiring", i.e. min-addition semiring over the reals. For two real numbers, "addition" of $x,y$ is defined as $\min(x,y)$, and "multiplication" of two reals is defined as $x + y$. With this definition of addition and multiplication, you can check that associativity holds for addition and multiplication, and that the distributive property holds, and that $0$ is the multiplicative identity, and multiplicative inverses exist (the inverse of $x$ is $-x$). However there is no additive identity and no additive inverses, because min is an "irreversible" operation. Hence why it's called a "semi-ring" instead of a ring. But it illustrates most of the ring properties and is very easy to understand and verify.

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The set of continuous functions $\mathcal{C}(\mathbb{R})$ from $\mathbb{R} $ to $\mathbb{R}$ is a ring under addition and multiplication defined as follows: $(f*g) (x) = f(x)g(x)$ and $(f+g)(x)=f(x)+g(x)$, this ring is also an interesting zoo of many interesting ring theoretic activities.

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The ring of $n \times n$ matrices, or $\Bbb R^n \longrightarrow \Bbb R^n$ linear maps, provides good examples of:

  • Nilpotents like $\left[\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}\right] (A^2 = 0 \text{ and } A \not= 0)$
  • Idempotent elements like $\left[\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}\right] (A^2 = A \text{ and } A \not\in \{0,I\})$
  • Multiplicative homomorphisms like $\det$
  • Noncommutativity

Look at the split-complex numbers and dual numbers to see good examples of ideals.

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