Let me quote the Wikipedia,
A group is a set, G, together with an operation $\cdot$ (called the
group law of G) that combines any two elements $a$ and $b$ to form
another element, denoted $a\cdot b$ or $ab$. To qualify as a group,
the set and operation, (G, $\cdot$), must satisfy four requirements
known as the group axioms: ....
Notice how they refer to a group as (G, $\cdot$). This emphasizes the fact that a group is precisely two things: A set G and an operation $\cdot$ which combines two elements of G together to form another element of the set G.
So, here are some examples of groups: $(\{0\}, +)$, $(\mathbb Z, +)$, $(\mathbb R, +)$, $(\mathbb R^n, +)$, $(\mathbb R\setminus\{0\}, \cdot)$, $(\mathbb R^+, \cdot)$ where $\mathbb R^+$is the set of positive real numbers, modular arithmetic $(\mathbb Z_n, +)$, $(\mathbb Z_p, \cdot)$ where $p$ is prime, the symmetry group of an equilateral triangle, the symmetry group of a square, and the Rubik's Cube Group. Notice that the group operation does not need to be denoted as a +. You can use any operation that takes two elements of G and maps them to G as long as that operation obeys the group axioms: closure, associativity, existence of an inverse, and an identity element.
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A Ring, on the other hand, is three things (R, +, $\cdot$) where R is a set, and + and $\cdot$ are two operations that combine two elements of R to form another element of R. As with groups, there are certain laws that the operations have to obey, for example the distributive law, the associative law, the operation + needs to obey the commutative law, ....
As you stated, you need to have a multiplication operation to form a Ring, so for example $(\mathbb Z, +)$ is not a Ring, but $(\mathbb Z, +, \cdot)$ is a Ring.
As you noticed, many Groups have a natural extension to Rings:
- The Group $(\mathbb Z, +)$ admits normal integer multiplication to form the ring $(\mathbb Z, +, \cdot)$.
- The Group $(\mathbb R, +)$ also admits multiplication to form the ring $(\mathbb R, +, \cdot)$.
- The Group $(\mathbb Z_n, +)$ admits modular multiplication to form the ring $(\mathbb Z_n, +, \cdot)$.
- The Group $(\mathbb R^+, \cdot)$ admits a multiplication operator to form the ring $(\mathbb R^+, \oplus, \odot)$ with $a\oplus b= a\cdot b$ and $a\odot b= \exp(\log(a)\log(b))$.
Some Groups do not admit a Ring structure. For details, see e.g. this post.
$(\mathbb R^3, +, \times)$ where $u\times v$ is the cross product of $u$ and $v$ is almost a ring, but it has no identity element for multiplication and $\times$ is not associative.
If the $\cdot$ in $(R,+,\cdot)$ is associative without an identity, then $(R,+,\cdot)$ is not a Ring. It is referred to as a Rng. Here are two nice examples from the Wikipedia:
A simple example of a rng that is not a ring is given by the even
integers with the ordinary addition and multiplication of integers.
Another example is given by the set of all 3-by-3 real matrices whose
bottom row is zero.
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You wrote "Also, the cyclic subgroup of order n. Am I right that it is not a ring since it is not a group under addition? This seems clear to me, but worried about stating this in class but I know that it is isomorphic to the set of integers modulo n that DOES constitute a ring.".
Every cyclic group is isomorphic to either $(Z, +)$ or $(Z_n, +)$. When you say the cyclic subgroup of order $n$, perhaps you are thinking of $(\{g^0, g^1, g^2, \ldots, g^{n-1}\}, \cdot)$ with $g^a\cdot g^b= g^{\mathrm{mod}(a+b,n)}$. You might think that this group cannot be extended to a Ring because the + operator is not defined. It can be extended to the Ring $(\{g^0, g^1, g^2, \ldots, g^{n-1}\}, \oplus, \odot)$ where $g^i\oplus g^j=g^{\mathrm{mod}(i+j,n)}$ and $g^i\odot g^j=g^{\mathrm{mod}(i\cdot j,n)}$.
If you have further questions, maybe you could put them in a comment below.
