Content about Ring - Abstract Algebra(2)

Concept:

Ring, Integral domain, Divison ring, field, unit, Matrix ring $R_n$ and base ring $R$, determinant, adjoint of the matrix, subring generated by S, center,ideal, difference ring, irreducible(prime), zero ring, characteristic, principle (one-side) ideal,

Denotation:

• subring generated by S: $[[S]]$
• subgroup generated by S: $[S]$

Definition:

If $a$ is an element of a ring $\mathfrak A$ for which there exists a $b\ne 0$ such that $ab=0$,then $a$ is called a left zero-divisor in $\mathfrak A$

因此 $0$ 显然是 zero-divisor.

• a ring is an integral domain if and only if it possesses no zero-divisors $\ne 0$

• a ring is an integral domain if and only if the restricted cancellation laws of multiplication hold.

• For ring, if the element identity exists, it is unique.

• If the identity 1=0, $\mathfrak A$ has only one element.

• Matrics $R_n$ with base ring $R$: 矩阵 $R_n$ 的环性质来自于底层集合 $R$ 的环性质

  • Multiplication of matrices is assosicative,i.e. $(ABC)_{ij}=\sum_{k,l}a_{ik}(b_{kl}c_{lj})$, thus the distributive laws hold.
  • Thus,the $R_n$ is a ring
  • but the commutative property is not inheritable. and it is not a integral domain.
  • if $R$ is a commutative ring with an identity, a matrix $(a)\in R_n$ is a unit if and only if its determinant is a unit in $R$
  • if $R=F$ is a field, a matrix $(a)\in F_n$ is a unit if and only if its determinant is different from zero.
    • $(a)adj(a)=[det(a)]_{diag}=adj(a)(a)$, $(a)\in R_n$
    • $det(a)(b)=det(a)det(b)$
    • R 为 commutative 是对行列式的伴随矩阵展开式进行操作的保障。

• the intersection of any collection of subring of a ring is a subring.

• if $S$ is a set of elements, the totality $C(S)$ of elements $c$ that commute with every $s\in S$ is a subring.

  • if $S_1 \supseteq S_2, C(S_1)\subseteq C(S_2)$
  • $C(C(S))\supseteq S$
  • $C(C(C(S)))=C(S)$
  • $C(S)=C([[S]])$

• A subset $\mathfrak B$ of a ring $\mathfrak A$ is called an ideal if $\mathfrak B$ is a subgroup of the additive group of $\mathfrak A$ and $\mathfrak B$ has the closure $(L)$ and $(R)$ property

  • $(L): ab\in \mathfrak B, \quad \forall a \in \mathfrak A \quad and \quad \forall b\in \mathfrak B$
  • $(R): ba\in \mathfrak B, \quad \forall a \in \mathfrak A \quad and \quad \forall b\in \mathfrak B$
  • 实际上，理想保证了商集合乘积运算的合理性
  • an ideal is closed under multiplication,such that it determines a subring of $\mathfrak A$

• the discussion in integer

  • $\mathfrak A$ can be an integral domain and have difference rings that are not integral domains.
  • $I/(m)$ 是一个典型的例子，其中 $I$ 是全体整数构成的集合。
  • $I/(p)$ is a field
    • the element $a$ is not divisible by $p$, such that $1=(a,p)$ and exists integers $b$ and $q$ that $ab+pq=1$
  • when $m$ is not a prime number, the units in $I/(m)$ have $(a,m)=1$
  • the order of the group M of units of $I/(m)$ is the number of positive integers that are less than m and are relatively prime to m
    • the order is denoted as $\phi(m)$
    • if $(a,m)=1$ ,then $a^{\phi(m)}=1$
    • Euler-Fermat: If a is an integer prime to the positive integer $m$, then
      $a^{\phi(m)}\equiv 1$ (mod m) .
    • Corollary: If $p$ is a prime and a $\not \equiv$ 0 (mod p), then $a ^{p-1}\equiv 1$ (mod p)

• Homomorphism of rings

  • if $\eta$ is a ring homomorphism of $\mathfrak A$ into $\mathfrak A'$, the image set $\mathfrak A \eta$ is a subring of $\mathfrak A'$
  • the kernel of a homomorphism of a ring $\mathfrak A$ is an ideal in $\mathfrak A$
  • discuss the ideal $\mathfrak B$ contained in the kernel $\mathcal R$, based on it, there is a ring homomorphism of the difference ring into the target ring. 当且仅当 $\mathfrak B$ 与 $\mathcal R$ 一样大时，这样的 ring homomorphism 是 isomorphism.
  • for any ring $\mathfrak A$ that has an identity $e$ and that is generated by $e$,consider the ring of integers $I$ and the mapping $n\rightarrow ne$ of $I$ into $\mathfrak A$. In fact, $1\rightarrow 1e=e$, so that $\mathfrak A$ is a homomorpgic image of $I$, it follows that $\mathfrak A \cong I/(m)$ where $m\ge 0$.
    • Thus either $\mathfrak A$ is infinite and isomorphic to the ring of integers or $\mathfrak A$ has a finite number $m$ of elements and $isomorphic$to the finite ring $I/(m)$

• the existence of zero ring

  • the definition of multiplication: $ab=0$

• A simple restrictions imposed on the multiplicative semi-group of a ring will impose strong restrictions on the additive group.

  • 从循环群的角度来看待下面的命题。
  • suppose that $\mathfrak A$ has an identity 1 and suppose 1 has finite order $m$ in $\mathfrak A,+$. Then for $a \in \mathfrak A$, $ma=0$, which means every element has finite order divisor of $m$ .
    • if there exists a maximum $m(>0)$ for the orders of the elements of $\mathfrak A,+$ then the number $m$ is called the characteristic of $\mathfrak A$ (if no such maximum exists, we say that $\mathcal A$ has characteristic 0 or infinity). so the characeristic varies if the order of identity is m or infinity.
  • suppose that $d$ is an element of $\mathfrak A$ that has finite order $m$ and $d$ is not a left zero-divisor.
    • for any $a\in \mathfrak A$, $ma=0$, meaning that the characteristic of $\mathfrak A$ is $m$.
    • if $\mathfrak A$ is an intergral domain of charateristic 0,then all of the non-zero elements of $\mathfrak A$ has infinite order. If $\mathfrak A$ has characteristic m>0, then $m$ is a prime and all of the non-zero elements of $\mathfrak A$ have order $m$

• Algebra of subgroups of the additive group of a ring

  • intersection
  • the group generated by a collection of subgroups
  • addition $[A+B]$
    • the additive group should be commutative.
  • products $AB$,$A^k$
    • a subgroup A of the additive group determines a subring if and only if A is closed under multiplication, meaning $A^2\subseteq A$
    • if $\mathfrak B$ is a subgroup such that (L) holds, then $\mathfrak B$ is a left ideal. (the right ideal like this)
    • In any ring $\mathfrak A$ the totality $\mathfrak A b$ of left multiples $xb$, $x in \mathfrak A$, is a left ideal. If $\mathfrak A$ contains an identity, then $\mathfrak A b$ contains $b$ and then $\mathfrak A b$ can be characterized as the smallest left ideal that contains $b$; If $\mathfrak A$ does not have an identity, it is necessary to take the set of elements of the form $nb+xb$, $n$ an integar, $x$ arbitrary in $\mathfrak A$, to obtain the smallest left ideal containing b. In any case we shall call the smallest left ideal containing an element $b$ a principal left ideal, denoted as $(b)_l$

• A ring $\mathfrak A$ with an identity $1 \ne 0$ is a division ring if and only if it has no proper left(right) ideals

  • the result implies that any division ring is simple. It follows that the only homomorphic images of a division ring are 0 and the ring itself.
  • the composition of intersection, sum and product applied to the left ideals give left ideals. For example, the product $\mathfrak {BE}$ is a left ideal if $\mathfrak B$ is any left ideal and $\mathfrak E$ is a subgroup(the multiplication). Also $\mathfrak {BE}$ is a two-sided ideal if $\mathfrak B$ is a left ideal and $\mathfrak E$ is a right ideal.

• The ring of endomorphisms of a commutative group

  • consider the set $\mathfrak E$ of endomorphisms of commutative additive group $\mathfrak G$, which are the mappings $\eta$ pf $\mathfrak G$ into itself such that $(a+b)\eta \rightarrow a\eta+b\eta$
  • the addition and multiplication is naturally definited.
  • Let $\mathfrak G$ be an arbitrary commutative group and let $\mathfrak E$ be the totality of endomorphism of $\mathfrak G$. Then $\mathfrak E$ is closed relative to the addition composition defined by $a(\eta+\rho)=a\eta+a\rho$ and relative to the resultant composition $\cdot$m and the system $\mathfrak E,+,\cdot$ is a ring.（这是重点）

• The multiplications of a ring

  • If a is a fixed element of $\mathfrak A$, we define the right multiplication $a_r$ to be the mapping $x\rightarrow xa$ of $\mathfrak A$ into itself.
  • the set $\mathfrak A_r$ of the right multiplications is a subring of all the endomorphisms $\mathfrak E$ (这是重点，是上一节的更具体讨论)
  • the kernel of the homomorphism $a\rightarrow a_r$ is th ideal $\mathfrak B_r$ of elements $z$ such that $xz=0$, which is called as the right annihialtor of the ring $\mathfrak A$ (这是相对于基环的称谓)
  • Any ring with an identity is isomorphic to a ring of endomorphisms (此时 $\mathfrak B_r=0$)
  • A similar discussion applies to the left multiplications $a_l$ defined by $x a_l=ax$. In fact, $a\rightarrow a_l$ is an anti-homomorphism
  • If $\mathfrak A$ is a ring with an identity, then any mapping in $\mathfrak A,+$ that commutes with all the left multiplications is a right multiplication. （整个证明基于identity的使用）