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?


\[n_R = 1 + \cdots + 1 \text{ (} n \text{ times)}\]

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 $R^\times$?


\[R^\times = \\{r \in R \mid \exists s \in R \text{ s.t. } r s = 1\\}\]

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.




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