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Ring homomorphisms

Some example computations with ideals and quotient rings

  • ideals of $\Z$ and quotients
  • ideals $(x^2+1)R$ of $R=\Q[x]$, $R=\R[x]$, and $R=\C[x]$
  • Ring of functions $X\to A$ and the evaluation map at points of $X$.
  • Evaluation map on polynomials $R[x]\to S$ extending $R\to S$ given by evaluation at $s\in S$.
  • $\Zn{p}[x]/(f(x))$
  • $\Z[i]/2$, $\Z[i]/3$, $\Z[i]/5$
  • two sided ideals of $M_{n}(R)$
  • examples of left ideals of $M_{n}(R)$.

Isomorphism theorems

  1. $R$ a ring, $A$ a subring, and $B$ and ideal of $R$. Let $A+B=\lbrace a+b | a\in A, b\in B\rbrace.$ Then $A\cap B$ is an ideal of $A$ and $(A+B)/B\isom A/(A\cap B)$.
  2. If $I$ and $J$ are ideals of $R$ and $I\subset J$, then $J/I$ is an ideal of $R/I$ and $(R/I)/(J/I)\isom R/J$.
  3. Let $I\subset R$ be an ideal. There is a bijective correspondence $A\to A/I$ between subrings of $R$ containing $I$ and subrings of $R/I$. This correspondence respects ideals, so $A/I$ is an ideal of $R/I$ if and only if $A$ is an ideal of $R$.

Sums and products of ideals

  1. The sum $I+J$ of two ideals is the collection of sums of elements of $I$ and $J$; it is an ideal.
  2. The product $IJ$ is the subring generated by products $ab$ with $a\in I$ and $b\in J$; it is an ideal.
  3. $I^{n}$ is the product of $I$ with itself $n$ times. it is an ideal.

Prime and maximal ideals

Suppose that $R$ has an identity element $1$ (and that $1$ is not zero, so the ring is not trivial).

  • An ideal $I=R$ if and only if $I$ contains a unit.
  • $R$ is a field iff its only ideals are zero and $R$.
  • Any homomorphism from a field $F$ to another ring $R$ is either zero or injective.