• Nonzero ring with:
  • The zero-product property: \( ab=0 \implies a=0\lor b=0 \), that is, no zero-divisor exists.
• It generally does not assume the commutativity, unlike 1.1.1.

Rng ⊃ Ring ⊃ Commutative Ring ⊃ 1.1.1 ⊃ Integrally Closed Domain ⊃ GCD Domain ⊃1.1.2 ⊃ 1.1.3 ⊃ 1.1.4 ⊃ Field ⊃ Algebraically Closed Field.

• A ring in which every non-zero element has a multiplicative inverse.
• This does not assume the commutativity of the multiplication.

• A ring \(R\) equipped finitely many derivations \(\Delta = \{\partial_i\}_{i=1}^n\) that satisfies, \(\forall r_1, r_2\in R\):
  • Distributivity: \(\partial_i (r_1 + r_2) = \partial_i r_1 + \partial_i r_2\)
  • Leibniz Rule: \(\partial_i (r_1r_2) = \partial_i(r_1) r_2 + r_1\partial_i (r_2)\)
  • Pairwise Commutativity: \(\partial_i\partial_j r = \partial_j\partial_i r\)

• A ideal \(I\) of a ring \((R, +, \cdot )\) is the additive subgroup of \(R\), that is absorbing under multiplication.
  1. \((I, +) < (R, +)\)
  2. \(\forall r\in R, \forall x\in I, rx\in I\) (in the case of left ideal)

• A radical \(\sqrt{I}\) of an ideal \(I\) in a commutative ring \(R\) is defined as: \[ \sqrt{I} := \{r\in R \mid \exists n\in \mathbb{Z}^+, r^n \in I\}. \]

• Invertible Element

• A nonzero, non-unit element \(a\in R\), such that \(\neg \exists r,s \in R, rs = a\), with non-unit \(r\) and \(s\).

• An element \(p \in R\), where for \(a, b\in R\), \(p\mid ab \implies (p\mid a \lor p\mid b)\).

• Two nonzero elements \(a,b\) such that: \[ ab = 0_R \]
• Zero-divisor does not have a multiplicative inverse.

• Smallest positive number \(n\) such that: \[ \underbrace{1+\cdots + 1}_{n} = 0 \]
• for the multiplicative identity \(1\) and the additive identity \(0\), if such a number exists, otherwise 0.
• Equivalently,
  • \(n\mathbb{Z}\) is the kernel of the unique ring homomorphism from \(\mathbb{Z}\) to \(R\).

• It is similar to the ./Group Theory.html#org3c5d3d0.
  • Any other element \(r\) in the ring is \(r\cdot 1\), therefore it also adds to the additive identity.
• If a nontrivial ring \(R\) does not have any nontrivial zero divisors, then its characteristic is either 0 or prime.
  • If the characteristic is a composite number, than the zero divisors are the repeated addition of the multiplicative identity by the factor times.
• If a commutative ring \(R\) has prime characteristic \(p\), then \[ \forall x, y\in R, (x+y)^p = x^p + y^p \quad\text{(Freshman's Dream)}. \]
  • The map \(x\mapsto x^p\) is a ring homomorphism \(R\to R\), called the Frobenius homomorphism.

• The quotient ring \(R/I\) consists of the equivalence classes of \(R\) with respect to the equivalence relation \(\sim\): \[ a\sim b \iff a-b\in I. \]
• The equivalence class of the element \(a\) is denoted by: \[ [a] = a+I = \lbrace a+r : r\in I\rbrace. \]
• By the definition of the ideal, the addition and multiplication is well-defined as:
  • \[ (a+I)+(b+I) = (a+b) + I, \]
  • \[ (a+I)(b+I) = (ab) + I. \]

• Every finite division ring is a field.
• That is, there is no distinction between finite domain, finite division ring, and finite field.

• \([L:K]\) is the dimension of the vector field \(L\) over \(K\), (hence \(L/K\)).

• Relative to a field extension \(L/K\), the minimal polynomial of \(\alpha\in L\) is the unique monic polynomial \(f(X) \in K[X]\) of least degree among the set of polynomials \(J_\alpha := \{f(X)\in K[X] \mid f(\alpha) = 0\}\)

• Splitting field of a polynomial \(f(X) \in K[X]\) is the smallest extension field of the field \(K\) in which the given polynomial \(f(X) \in K[X]\) splits into linear factors.

• A field extension \(L/K\) is algebraic if for all \(\alpha\in L\) it has nonzero polynomial \(f(X) \in K[X]\) such that \(f(\alpha) = 0\).

• An algebraic field extension \(L/K\) is separable if for all \(\alpha\in L\), the minimal polynomial of \(\alpha\) over \(K\) is a .

• An algebraic field extension \(L/K\) is called a normal extension if every irreducible polynomial over \(K\) that has a root in \(L\) splits into linear factors in \(L\).

• An algebraic field extension \(L/K\) that is normal and separable.

• A field extension \(L/K\) where: for a integer \(n >1\),
  • \(K\) contains \(n\) distinct \(n\)th roots of unity
  • \(L/K\) has abelian Galois group of exponent \(n\)

• A field obtained by adjoining a complex root of unity to \(\mathbb{Q}\)

• A field \(K\) is algebraically closed if for all \(\alpha \in K\) there exists \(f(x)\in K[X]\) such that \(f(\alpha) = 0\).

• An algebraic closure \(\overline{K}\) of a field \(K\) is an algebraic extension of \(K\) that is algebraically closed.

• Noncommutative polynomial ring

• For a field \(K\) and its multiplicative group \(K^\times\), and an abelian totally ordered group \((\Gamma, +, \ge)\) with the ordering on \(\Gamma\) extended to \(\Gamma\cup \{\infty\}\) by the rule: \(\forall a \in \Gamma, \infty \ge a\) and \(\infty + a = a+ \infty = \infty + \infty = \infty\),
• The valuation of \(K\) is any map \[ \nu\colon K \to \Gamma \cup\{\infty\} \] that satisfies:
  • \(\nu(a) = \infty \iff a = 0\)
  • \(\nu(ab) = \nu(a) + \nu(b)\)
  • \(\nu(a+ b) \ge \min(\nu(a), \nu(b))\), with equality if \(\nu(a)\neq \nu(b)\)