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| author | JJ | 2024-02-21 01:28:49 +0000 |
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| committer | JJ | 2024-02-21 01:28:49 +0000 |
| commit | fa1e6f75ceb4f2a99dc1c107bfe60434356cfe19 (patch) | |
| tree | 2c76452361bdbe2166b24f8dd665ac51b4e26ca7 /mathematics/algebra.md | |
| parent | 19ab14173c4242792cee0bd9b75d81c788e0fb70 (diff) | |
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| -rw-r--r-- | mathematics/algebra.md | 30 |
1 files changed, 11 insertions, 19 deletions
diff --git a/mathematics/algebra.md b/mathematics/algebra.md index 4b9a097..3756604 100644 --- a/mathematics/algebra.md +++ b/mathematics/algebra.md @@ -5,8 +5,8 @@ title: mathematics/algebra # algebra -modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, vector spaces, lattices, and the like. -these structures are very general: and so results from abstract algebra can be applied to a wide variety of situations. +modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, and the like. +these structures are very abstract: and so results can be applied to a wide variety of situations. ## structures @@ -34,9 +34,9 @@ A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the f - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$ - $∀a,b,c : a × (b+c) = (a×b) + (a×c)$ - An Abelian or **commutative ring** satisfies an additional axiom: - - commutativity (of ×): $∀a,b : a×b=b×a$ + - commutativity (of $×$): $∀a,b : a×b=b×a$ -A **field** is a *commutative ring* where $0 ≠ 1$ and all elements sans $0$ have an inverse $a^{-1}$ under multiplication. +A **field** is a *commutative ring* where all elements sans $0$ have an inverse $a^{-1}$ under multiplication. Subsequently, $0 ≠ 1$. A field may be also thought of as a set on which addition, subtraction, multiplication, and division are defined and behave as they do on $ℝ$. A **vector space** $V$ over a field $F$ of scalars is a set with a binary operation $+$ and a binary function satisfying the following axioms: - $(V, +)$ is a *commutative group*: @@ -53,7 +53,7 @@ A **vector space** $V$ over a field $F$ of scalars is a set with a binary operat A **module** $M$ is a generalization of a *vector space* to function over a ring $R$ instead of a field. -A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the following axioms: +A **lattice** $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms: - commutativity: - $∀a,b : a ∧ b = b ∧ a$ - $∀a,b : a ∨ b = b ∨ a$ @@ -67,22 +67,14 @@ A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the - $∀a : a ∧ a = a$ - $∀a : a ∨ a = a$ -## group theory +## [group theory](group-theory.md) -... +## [ring theory](ring-theory.md) -## ring theory +## [galois theory](galois-theory.md) -... +## [linear algebra](linear-algebra.md) -## galois theory +## [order theory](order-theory.md) -... - -## linear algebra - -... - -## order theory - -a lattice may alternatively be defined as... +## [coding theory](coding-theory.md) |