ISLAM is one of the derivatives of a common root, S-L-M, for many related words in Arabic such as AS-SALAM and MUSLIM. The derivatives are formed by applying the root form to different linguistic patterns.
The root SLM itself literally means “peace.” When the first consonant of the root is prefixed with the patternal “i-,” i.e., iSLM and the last with “a-,” i.e. SlaM, altogether the root reads iSLaM now. This is the pattern of full action. Thus, Islam means "putting peace into action" or "practicing peace fully." So, in reality the word iSLaM refers to the application of peace not just peace itself. This is why Islam cannot be without practice.
The word muSLiM is yet another derivative of the same root, SLM, through addition of a different set, i.e., “mu-” before the S, and “i-” after the L. The equivalent of this pattern in English is the suffex -er like in read-er, do-er, or act-or, etc. So, derived from iSLaM, the practicing of peace, the word muSLiM now means one who is practicing iSLaM, practicer of peace, a peaceful person or a peace-maker. When you add the suffix -ah, you generate female form of the doer. Thus, muSLiMah is the female practicer of peace.
When you add the prefix al-, it functions as a definite article. Hence, Al salam is the peace. If "As salamu alaykum" literally is “The peace upon you," then alaykum must mean the rest of the translation which is “Upon you.” Yes, “ala” in Arabic on, upon, and “kum” is you (plural).
The Hebrew words shalom and Solomon are also peace-related words in meaning as Hebrew and Arabic are sister languages.