TY - JOUR T1 - Radical and It’s Applications in BCH-Algebras TT - JF - IJMSI JO - IJMSI VL - 8 IS - 1 UR - http://ijmsi.ir/article-1-399-en.html Y1 - 2013 SP - 15 EP - 29 KW - Ideal KW - radical KW - Quotient $BCH$-algebra KW - Maximal KW - Translation. N2 - Let $X$ be a $BCH$-algebra and $I$ be an ideal of $X$. In this paper, we introduce the concept of $sqrt{I}$. We show that it is an ideal of $X$, when $I$ is closed ideal of $X$. Then we verify some useful properties of it. We prove that it is the ::::union:::: of all $k-$nil ideals of $I$. Moreover, if $I$ is a closed ideal of $X$, then $sqrt{I}$ is a closed translation ideal and so we can construct a quotient $BCH$-algebra. We prove this quotient is a P-semisimple $BCI$-algebra and so it is an abelian group. Then we use the concept of radical in order to construct the second and the third isomorphism theorems. M3 10.7508/ijmsi.2013.01.002 ER -