📂Abstract Algebra

Rings in Abstract Algebra

Definition ¹

A set $R$ satisfying the following rules for two binary operations, addition$+$ and multiplication$\cdot$, is defined as a Ring.

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When $a$, $b$, $c$ are elements of $R$,

1. Commutative law holds for addition. $$a+b=b+a$$
2. Associative law holds for addition. $$(a+b)+c=a+(b+c)$$
3. There exists an identity element for addition. $$\forall a \ \exists 0\ \ \mathrm{s.t} \ a+0=a$$
4. There exists an additive inverse for every element. $$\forall a \ \exists -a\ \ \mathrm{s.t}\ a+(-a)=0$$
5. Associative law holds for multiplication. $$(ab)c=a(bc)$$
6. Distributive law holds for addition and multiplication. $$a(b+c)=ab+ac\ \mathrm{and} \ (b+c)a=ba+ca$$

Description

In summary, a set $R$ is called a ring when it is an Abelian group under addition and a semigroup under multiplication, and the distributive law is applicable for both operations.

Specifically, if the commutative law also holds for multiplication, it is called a commutative ring or Abelian ring. Moreover, according to the definition of a ring, there’s no need for the existence of an identity or an inverse for multiplication. Even if an identity exists, an inverse is not necessary. A set is considered a ring if it satisfies the six conditions mentioned above.

When dealing with groups, we denote the identity element for an operation as $e$. Since there are two operations in a ring, different symbols are used to easily identify the identity for each operation. The identity for addition is denoted as $0$ and called the identity. If an identity for multiplication exists, it is denoted as $1$ and called unity. An element $a$ with an existing multiplicative inverse in the ring $R$ is referred to as a unit.

Similar to groups, the existence of a multiplicative identity in a ring, if it exists, is unique. Likewise, if an element’s inverse exists, it is also unique. The proof of this is identical to that used for groups, so it will not be reiterated here. Please refer to this link for more details.

Example

Consider the set of integers $\mathbb{Z}$. It satisfies the six conditions mentioned, making it a ring under addition and multiplication. It is also a commutative ring since the commutative law is satisfied for multiplication. The unity $1$ exists, and its element is the integer 1, with units being 1 and -1 (each having 1 and -1 as their inverses, respectively).

Note

In a ring, the existence of a multiplicative identity and inverse is not ’necessary’. Hence, one cannot indiscriminately cancel like in a group. What this means is when $a,\ b,\ c$ is an element of the ring $R$, just because $ab=ac$, one cannot hastily conclude that $b=c$. This is because the inverse for $a$ does not necessarily exist.

Likewise, $a^2=a$ does not warrant hastily drawing the conclusions that $a=0$ or $a=1$. This is an important consideration when dealing with rings.