By Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko

Accosiative jewelry and algebras are very attention-grabbing algebraic constructions. In a strict experience, the idea of algebras (in specific, noncommutative algebras) originated fromasingleexample, namelythequaternions, createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative"numbersystem." in the course of thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, started to deliver a few order into them and to unmarried out specific sorts of algebras for specified consciousness. hence, low-dimensional algebras, department algebras, and commutative algebras, have been classi?ed and characterised. The ?rst entire ends up in the constitution idea of associative algebras over the genuine and complicated ?elds have been bought via T.Molien, E.Cartan and G.Frobenius. smooth ring concept all started whilst J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. 20 years later, E.Artin proved a constitution theorem for earrings pleasant either the ascending and descending chain which generalized Wedderburn constitution theorem. The Wedderburn-Artin theorem has on the grounds that develop into a corn- stone of noncommutative ring idea. the aim of this publication is to introduce the topic of the constitution thought of associative earrings. This booklet is addressed to a reader who needs to benefit this subject from the start to investigate point. we've attempted to put in writing a self-contained publication that's meant to be a latest textbook at the constitution thought of associative jewelry and similar buildings and may be obtainable for autonomous research.

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**Example text**

J∈J (c) ⇒ (a). We shall show that any nonzero submodule N of an A-module M contains a simple submodule. Let n ∈ N and n = 0. The kernel of the homomorphism A → nA, for which a → na, is a right ideal X in the ring A. 3, there exists a maximal right ideal Y of A such that X ⊂ Y . Then Y /X is a maximal submodule in A/X. Therefore nY is a maximal submodule in nA. Then, by assumption, there exists a submodule M of M such that M = nY ⊕ M . Let na ∈ nA, then na = ny + m , where m ∈ M . This gives a direct decomposition nA = nY ⊕ (M ∩ nA) because m ∈ nA.

Let L be a submodule of M and π : M → M/L be the natural projection. For any submodule N ⊂ M and any submodule N ⊂ M/L we have 1) π(N ) is a submodule of M/L; 2) π −1 (N ) is a submodule of M ; 3) π(π −1 (N )) = N ; 4) if L ⊂ N then π −1 (π(N )) = N . As a corollary of this lemma we have the following theorem. 5 (Second isomorphism theorem). Let L be a submodule of an A-module M . Then any submodule of the A-module M/L has the form N/L, where L ⊂ N ⊂ M , and (M/L)/(N/L) M/N. Proof. Let π : M → M/L be the natural projection.

Am } ∈ P(S) can be written as A = {a1 } ∪ {a2 } ∪ ... ∪ {am }. 7. The Boolean algebra B has a unique atom which is equal to 1. 8. The Boolean algebra Bn has n atoms. 0) with 1 at the i-th position and 0 elsewhere. , bn ) ∈ Bn can be written as b = ei1 ∨ ei2 ∨ ... , k and bij = 0 for other ij . 8. A nonzero element a of a Boolean algebra B is an atom if and only if the inequality x ≤ a has exactly two solutions x = a and x = 0. Proof. Let a nonzero element a ∈ B be an atom, and suppose x ≤ a, where a = 0.