Notes - Rings and Modules HT24, Basic definitions for rings
Most definitions covered in [[Notes - Linear Algebra MT23, Rings]]U and [[Notes - Linear Algebra MT23, Fields]]U.
Flashcards
Suppose:
- $R$ is a ring
- $S$ is a subset of $R$
Can you state the subring criterion for $S$?
$S$ is a subring iff $1 \in S$ and $\forall s _ 1, s _ 2 \in S$:
- $s _ 1s _ 2 \in S$
- $s _ 1 - s _ 2 \in S$
Suppose:
- $R$ is a ring
- $S$ is a subset of $R$
What are the three conditions in the subring criterion?
- $1 \in S$
- $s _ 1 s _ 2 \in S$ $\forall s _ 1, s _ 2 \in S$:
- $s _ 1 - s _ 2 \in S$ $\forall s _ 1, s _ 2 \in S$:
Suppose:
- $R$ is a ring
- $n \in \mathbb Z$
How is $n _ R$ defined?
Suppose:
- $R$ is a ring
- $n \in \mathbb Z$
- $n _ R = 1 + \cdots + 1 \text{ (} n \text{ times)}$
- $\phi : \mathbb Z \to R$ given by $n \mapsto n _ R$
Why is $\text{Im } \phi$ the smallest subring of $R$?
Any subring contains the identity, and is closed under addition.
Suppose:
- $R$ is a ring
- $n \in \mathbb Z$
- $n _ R = 1 + \cdots + 1 \text{ (} n \text{ times)}$
- $\phi : \mathbb Z \to R$ given by $n \mapsto n _ R$
How is the characteristic of the ring defined informally and then formally?
Informally: the characteristic is $0$ if $n _ R \ne 0$ for all $n$, and $n$ if $n _ R$ is the smallest $n$ such that $n _ R = 0$.
Formally: by the first isomorphism theorem, $\mathbb Z / \ker \phi$ is isomorphic to $\text{Im } \phi$, which is a cyclic group. Then $\mathbb Z / \ker \phi = \mathbb Z / d\mathbb Z$ for some $d \ge 0$, since every finite cyclic group is isomorphic to $\mathbb Z / k\mathbb Z$ for some $k \ge 1$ and every infinite cyclic group is isomorphic to $\mathbb Z = \mathbb Z / 0\mathbb Z$. In this case, $d$ is the characteristic.
What is a unit in a ring?
An elemnt with a multiplicative inverse.
Suppose $R$ is a ring. Can you define $R^\times$?
i.e. the group of units of $R$.
What does it mean for a function $\phi : R \to S$ to be an embedding?
It is an injective homomorphism.
Suppose:
- $R$ is a ring
- $a, b \in R$
What does it mean to say that $a$ and $b$ are associates?
There exists a unit $u \in \mathbb R^\times$ such that $a = ub$.
What does it mean for some non-unit non-zero element $r$ of an integral domain $R$ to be irreducible?
Whenever $r = ab$, then exactly one of $a$ or $b$ is a unit (if both were, then $r$ would be a unit).
In the definition of an irreducible or prime element $r \in R$, what two cases are excluded by definition?
$r$ is not a unit, and non-zero.
If $R$ is a ring, what does the notation $U(R)$ mean?
The group of units in $R$.
What is a unital ring?
A ring with a multiplicative identity.