Download Boolean algebra by R. L. Goodstein PDF

By R. L. Goodstein

Show description

Read or Download Boolean algebra PDF

Best algebra books

Finite Semigroups and Universal Algebra (Series in Algebra, Vol 3)

Prompted through functions in theoretical computing device technology, the speculation of finite semigroups has emerged in recent times as an self sufficient sector of arithmetic. It fruitfully combines equipment, rules and buildings from algebra, combinatorics, common sense and topology. merely, the speculation goals at a category of finite semigroups in yes periods known as "pseudovarieties".

Extra resources for Boolean algebra

Example text

2: Let R be a commutative ring with 1, and n ≥ 1 be an integer. Mn×n = {(aij) / aij ∈ R; 1 ≤ i, j ≤ n} be the set of all n × n matrices with entries in R where  a11 a12 . . a1n     a 21 a 22 . . a 2 n   . .    (aij ) = . .    . .    a  n1 an 2 . . ann  Define addition and multiplication in Mn×n as follows: (aij) + (bij) = (aij + bij) n and (aij ) . (bij ) = (cij ) where cij = ∑ aik bkj for all i, j; 1 ≤ i, j ≤ n. It is easily k=1 verified that Mn×n is a ring called the matrix ring of order with entries from R.

Let S = {0, 1 + p1 + p2 + p3 + p4 + p5} be the subset of Z2S3. S is a S-pseudo ideal related to A and S is also a S-pseudo ideal related to Z2. 6: Let Z2 = {0, 1} be the prime field of characteristic 2, G any finite group of order n. Then Z2G has S-pseudo ideals which are ideals of Z2G. Proof: Take Z2 = {0, 1} a field of characteristic two and the group ring Z2G is a S-ring. Let G = {g1, g2, …, gn–1, 1} be the set of all elements of G. S = {0, 1 + g1 + … + gn–1} is a S-pseudo ideal related to Z2 and S is also an ideal of Z2G.

2: Let R be a ring say Z8. Now M = Z8 × Z8 is an abelian group under addition, M is a Z8 – module over Z8. 4: Let M be a module, M is called a simple module if M ≠ (0) and the only submodules of M are (0) and M. 3: Let R be a ring S = R × R is an R-module. (show M = R × {0} and N = {0} × R are not isomorphic as S-modules). PROBLEMS: 1. Let A and B be two submodules of a module M; prove A ∩ B is a submodule of M. 2. M = Z × Z × Z × Z × Z is a module over Z. 1. Find submodules of M. 2. Find two submodules which are isomorphic in M.

Download PDF sample

Boolean algebra by R. L. Goodstein


by Anthony
4.2

Rated 4.18 of 5 – based on 21 votes