By Dan Laksov

**Read or Download Algebra PDF**

**Similar elementary books**

**Living Dairy-Free For Dummies (For Dummies (Health & Fitness))**

Regardless of the cause of no longer consuming dairy, dwelling Dairy-Free For Dummies offers readers with the main up to date details on a dairy-free nutrition and way of life and may empower them to thrive with out dairy whereas nonetheless getting the calcium, diet D and dietary advantages normally linked to dairy items.

**Beginning and Intermediate Algebra, 3rd Edition **

Development a greater route to luck! Connecting wisdom – Sherri prepares her scholars for fulfillment by way of clean their wisdom of mathematics. by way of supporting scholars see the relationship among mathematics and algebra, Sherri discovered that her scholars have been extra convinced of their skills as they stepped forward in the course of the direction.

- Komplexe Zahlen Fur Dummies Das Pocketbuch
- Functions and Graphs
- Algebra, Edition: 3., überarb. u. erw. Aufl.
- Ext GWT 2.0: Beginner's Guide
- The Nature of Mathematics

**Additional info for Algebra**

**Example text**

Let m be the maximal ideal pZ(p) of Z(p) . Show that the residue ring Z(p) /m is canonically isomorphic to Z/(p). 7. Let A[t] be the polynomial ring in the variable t with coefficient in A and let p be a prime ideal in A. Show that the set pA[t] of all polynomials with coefficietns in p form a prime ideal in A[t]. 8. Let K[u, v] be the polynomial ring in the two variables u and v over the field K. Is the union (u) ∪ (v) of the two ideals (u) and (v) of K[u, v] an ideal? 9. Show that if A is a ring such that 1 = 0 then A has minimal prime ideals.

2) When D(f ) ⊆ ∪α∈I D(fα ) we have that f ∈ r(a). Consequently there is a positive integer n and a finite subset J of I such that f n = β∈J fβ hβ with hβ ∈ A for all β in a finite set J . Hence f ∈ r((fβ )β∈J ) and it follows from assertion (1) n that D(f ) ⊆ ∪β∈J D(fβ ). However D(fβ ) = D(fβ β ) for all positive integers nβ . n Consequently D(f ) ⊆ ∪β∈J D(fβ β ), and using assertion (1) once more we obtain the n inclusion f ∈ r((fβ β )β∈J ) which is equivalent to the equality of assertion (2).

On the other hand we have that f1 f2 · · · fn ∈ / p, n which contradicts the assumption that ∩i=1 ai ⊆ p. 23) Exercises. 1. Let K be a field and let K[t] be the polynomial ring in the variable t over K. (1) Find all non-zero divisors in the residue class ring k[t]/(t2 ). (2) Find all the units in the residue class ring k[t]/(t2 ). 2. Let n be a positive integer. (1) Determine for which integers n the ring Z/nZ is an integral domain. (2) Determine for which integers n the ring Z/nZ is a field. 3.