د. حسن محمد علي سعيد شلاكه الاسدي

An infinite dimensional associative algebra ${\mathcal{A}}$ ${\mathcal{A}}$ over a field ${\mathbb{F}}$ is called locally finite associative algebra if every finite set of elements is contained in a finite dimensional subalgebra of ${\mathcal{A}}$. Given any associative algebra ${\mathcal{A}}$ over field ${\mathbb{F}}$ of any characteristic. Consider a new multiplication on ${\mathcal{A}}$ called the Lie multiplication which defined by [a, b] = ab − ba for all a, b ∈ A, where ab is the associative multiplication in ${\mathcal{A}}$. Then L = ${\mathcal{A}}$(−) together with the Lie multiplication form a Lie subalgebra of ${\mathcal{A}}$. It is natural to expect that the structures of L and ${\mathcal{A}}$ are connected closely. In this paper, we study and discuss the structure of infinite dimensional locally finite Lie and associative algebras. The relation between them, their ideals and their inner ideals is considered. A brief discussion of the simple associative algebras and simple Lie algebras is also be provided.

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

An infinite dimensional associative algebra ${\mathcal{A}}$ ${\mathcal{A}}$ over a field ${\mathbb{F}}$ is called locally finite associative algebra if every finite set of elements is contained in a finite dimensional subalgebra of ${\mathcal{A}}$. Given any associative algebra ${\mathcal{A}}$ over field ${\mathbb{F}}$ of any characteristic. Consider a new multiplication on ${\mathcal{A}}$ called the Lie multiplication which defined by [a, b] = ab − ba for all a, b ∈ A, where ab is the associative multiplication in ${\mathcal{A}}$. Then L = ${\mathcal{A}}$(−) together with the Lie multiplication form a Lie subalgebra of ${\mathcal{A}}$. It is natural to expect that the structures of L and ${\mathcal{A}}$ are connected closely. In this paper, we study and discuss the structure of infinite dimensional locally finite Lie and associative algebras. The relation between them, their ideals and their inner ideals is considered. A brief discussion of the simple associative algebras and simple Lie algebras is also be provided.

A subspace B of a Lie algebra L is said to be an inner ideal if [B, [B,L]] ⊆ B. Suppose that L is a Lie subalgebra of an associative algebra A. Then an inner ideal B of L is said to be Jordan-Lie if B2 = 0. In this thesis, we study Jordan-Lie inner ideals of finite dimensional associative algebras (with involution) and their corresponding Lie algebras over an algebraically closed field F of characteristic not 2 or 3. Let A be a finite dimensional associative algebra over F. Recall that A becomes a Lie algebra A(-) under the Lie bracket defined by [x,y] = xy - yx for all x,y ∈ A. Put A(0) = A(-) and A(k) = [A(k-1),A(k-1)] for all k ≥ 1. Let L be the Lie algebra A(k) (k ≥ 0). In the first half of this thesis, we prove that every Jordan-Lie inner ideal of L admits Levi decomposition. We get full classification of Jordan-Lie inner ideals of L satisfying a certain minimality condition. In the second half of this thesis, we study Jordan-Lie inner ideals of Lie subalgebras of finite dimensional associative algebras with involution. Let A be a finite dimensional associative algebra over F with involution * and let K(1) be the derived Lie subalgebra of the Lie algebra K of the skew-symmetric elements of A with respect to *. We classify * -regular inner ideals of K and K(1) satisfying a certain minimality condition and show that every bar-minimal * -regular inner ideal of K or K(1) is of the form eKe* for some idempotent e in A with e*e = 0. Finally, we study Jordan-Lie inner ideals of K(1) in the case when A does not have “small” quotients and show that they admit *-invariant Levi decomposition.

Let A be a finite dimensional associative algebra over a perfect field and let R be the radical of A. We show that for every one-sided ideal I of A there exists a semisimple subalgebra S of A such that I=IS⊕IR where IS = I∩S and IR = I∩R.

In this paper, we generalize some basic concepts to a BH-algebra, we find the relation between the notion of an associative BH-algebra and a Group of abstract algebra and we define the notion of a closed ideal with respect to an element of a BH-algebra which is a generalization of the notion of a closed ideal with respect to an element of a BCH-algebra.

In this paper, we define the concepts of a fuzzy closed ideal with respect to an element of a BH-algebra and a fuzzy closed BH-algebra with respect to an element of BH-algebra. We stated and proved some theorems which determine the relation between these notion and the notions of fuzzy ideals in a BH-algebra.

In this paper, we define the concepts of a closed ideal with respect to an element of a BCH-algebra and a closed BCH-algebra with respect to an element of BCH-algebra . We stated and proved some theorems which determine the relationship between these notion and the notions of some ideals of a BCH-algebra.