Additive Subgroup

Definition

$$H \subseteq G \text{ is an additive subgroup of } (G,+) \iff \forall a,b \in H: a+b \in H \text{ and } -a \in H.$$

In abstract algebra, many structures are built upon an underlying set with an operation called addition. An additive subgroup captures the idea of a smaller structure within an additive group that itself forms a group under the same operation. This concept is fundamental for understanding more complex structures like ideals in ring theory, where ideals are additive subgroups closed under multiplication by ring elements.

Notes

We begin by recalling the notion of a group. A group is a set $G$ together with a binary operation $\cdot$ (often denoted as $+$ for abelian groups) satisfying: closure, associativity, existence of identity, and existence of inverses. When the operation is written as $+$, the group is often called an additive group and is usually abelian (commutative). The additive identity is $0$, and the additive inverse of $a$ is $-a$.

A subgroup of a group is a subset $H \subseteq G$ that is itself a group under the same operation. For an additive group $(G,+)$, an additive subgroup is a subgroup in this sense. Equivalently, $H$ is an additive subgroup if it is nonempty, closed under addition: $a,b \in H \Rightarrow a+b \in H$, and closed under taking additive inverses: $a \in H \Rightarrow -a \in H$. Note that closure under addition and inverses automatically implies the subgroup contains the identity $0$ (since $0 = a + (-a)$).

Additive subgroups are the basic building blocks for rings: the additive group of a ring is an abelian group, and its additive subgroups are the first step toward defining subrings and ideals. An ideal in a ring is an additive subgroup that also satisfies $ra \in I$ and $ar \in I$ for all $r$ in the ring and $a$ in $I$. Thus, additive subgroups are prerequisites for ideal theory.

Examples: In $\mathbb{Z}$ under addition, the set $n\mathbb{Z} = \{ nk \mid k\in\mathbb{Z} \}$ is an additive subgroup for any integer $n$. In $\mathbb{R}^2$ under vector addition, any line through the origin is an additive subgroup.

Relation to Other Structures

Additive subgroups appear in various algebraic contexts. In module theory, a submodule of an $R$-module is an additive subgroup closed under scalar multiplication. In linear algebra, a subspace of a vector space is an additive subgroup closed under scalar multiplication.

Examples

1. Even integers: $2\mathbb{Z}$ is an additive subgroup of $\mathbb{Z}$ because sum of two evens is even, and the negative of an even is even.
2. Rational numbers: $\mathbb{Q}$ is an additive subgroup of $\mathbb{R}$.
3. Diagonal matrices: The set of $2\times2$ diagonal matrices with real entries is an additive subgroup of $M_2(\mathbb{R})$ under matrix addition.

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Significance

Foundation for Ring Theory

Additive subgroups are the underlying additive structure for subrings and ideals. Understanding them is essential for studying the decomposition of rings and modules.

General Algebra

The concept of subgroup is one of the first abstract algebraic structures, and additive subgroups provide a concrete intuition for the more general notion of subgroups in group theory.

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Glossary

Term / Symbol	Meaning
Additive group	A group whose operation is denoted by addition, typically abelian.
Subgroup	A subset of a group that is itself a group under the same operation.
Additive identity	The element 0 such that a+0 = a for all a in the group.
Additive inverse	For an element a, the element -a such that a + (-a) = 0.
Closure	The property that the sum of two elements in the subset is also in the subset.