Algebra I - Commutative Rings

A subset S of a ring R is a subring of R if

- 1 is element of S

- if a,b are in S, then a+b is in S

- if a,b are in S, then ab is in S

A ring R is commutative, if ab = ba for all a and b in R

A domain (or integral domain) is a commutative ring R that satisifies two extra axioms:

- 1 is not equal to zero

- Cancellation Law (for all a,b in R: if ca = cb and c not zero then a = b)

\(\mathbb{Z}_m\)  is a domain, if and only of m is prime

An element u in a commutative Ring R is called a unit, if u | 1 in R, that is if there exists an v in R with uv = 1

The element v is called the multiplicative inverse of u and denotetd with u^-1

A field F is a commutative ring in which \(1 \neq 0\) and every nonzero element a is a unit. that is, there is a a^-1  in F with a^-1a=1.

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The field F constructed from R (the set of all equivalence classes of (a,b) by (a,b) = (c,d) if ad = bc) is called the fractionfield of R, denoted by: Frac(R)

if k is a field, then every polynomial f(x) in k[x] of degree >= 1 is a product of a nonzero constant and monic irreducibles and

\(f(x) = ap_1(x) \dots p_m(x)\)

this factorization is unique bis auf eine umsortierung der Reihenfolge.....

An element p in a domain R  is irreducible if p is not zero and p is no unitg and there exist no none units u,v such that

p = uv

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Zu deutsch: Hauptideal

In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R.

If k is a field an f,g in k[x] with f not zero, then there are unigque polynomials q and r sucht that:

g  = qf + r

wehtere either r = 0 or deg(r) < deg(f)

The indeterminate x in R[x] is

\(x = (0,1,0,0,\dots)\)

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\(\sigma + \tau = (s_0 + t_0, s_1 + t_1, \dots , s_n + t_n, \dots)\)

\(\sigma \tau = (c_0, c_1, \dots)\)

with \(c_k = \sum_{i = 0}^k s_i\cdot t_{k-i}\) wehre k = i+j

A polynomial over a commutative ring R is a formal power series

\(\sigma = (s_0,s_1,\dots,s_n,0,\dots,0,\dots)\)

over R for which there exists some integer \(n \geq0\) with \(s_i = 0 \text{ for all} i > n\)

So a polynomial has only finetly many nonzero coefficients.

We call \(s_n\) the leading coefficient of \(\sigma \) and we denote the degree by \(deg(\sigma) = n\)

If R is a commutative ring, then a formal power series over R is a sequence of elements \(s_i \in R \text{ for all } i\geq 0\) called the coefficients of \(\sigma \):

\(\sigma = (s_0, s_1, \dots ,s_i, \dots)\)

Let R be a commutative ring an I a subset of R which is a ideal

I is prime ideal if and only of R/I is a integral domain

Let R be a commutative ring and I a subset of R which is an ideal.

I is maximal if and only if R/I is a field

Let R and A be commutative rings. If \(\phi: R \rightarrow A \) is a homomorphism, then \(ker(\phi)\) is an deal in R and \(im(\phi)\) is a subring of A and it satisfy

\(R / ker(\phi) \cong im(\phi)\)

Let I be an ideal in a commutative ring R. The natural map or projection is the function:

\(\pi: R \rightarrow R/I, x \rightarrow [x]\)

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The ring constructed trough the "Congruence Mod I" equivalence relation is called the Quotient Ring of R modulo I and is denoted by: R/I, so it is the set of all equivalent classes.......

Given is an ideal I in a commutative Ring R. The relation ~on R, defined by:

\(a \sim b \Leftrightarrow a-b \in I\)

and it is a equivalence relation on R

If \(b_1,b_2,\dots,b_n\) lie in R, then the set of all linear combinations:

\(I = \{r_1b_1 + r_2b_2 + \dots + r_nb_n \quad | \quad r_i \in R \quad \forall i\}\)

is an ideal in R an we write \((b_1, \dots , b_n)R\)

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An ideal in a commutative ring R is a subset I of R such that :

\(0 \in I\)

\(a,b \in I \Rightarrow a+b \in I\)

\(a \in I, b\in R \Rightarrow ab \in I\)

\(ker(\phi) := \{a \in A \quad | \quad \phi(a) = 0 \}\)

\(im(\phi) := \{\phi(a) \quad | \quad a \in A \}\)

If A,B  are rings, a ring homomorphism is a function \(\phi: A \rightarrow B\) such that:

\(\phi(1) = 1\)

\(\phi(a + b) = \phi(a) + \phi(b), \quad a,b \in A\)