Barisan $R$-modul $ A \xlongrightarrow{f} B \xlongrightarrow{g} C$ dikatakan $U$-eksak di $B$ jika terda-pat submodul $U$ di $C$ \ sedemikian hingga $Im(f) = g^{-1} (U)$. Barisan $R$-modul $ A \xlongrightarrow{f} B \xlongrightarrow{g} C$ dikatakan $V$-koeksak (di $B$) jika terdapat submodul $V$ di $A$ \ sedemikian hingga $f(V) = Ker(g)$. Barisan $V$-koeksak merupakan dual barisan $U$-eksak. Barisan $U$-eksak pendek dengan prapeta dari $U$ merupakan penjumlah langsung disebut barisan $U$-terpisah, sedangkan barisan $V$-koeksak pendek dengan peta $V$ merupakan penjumlah langsung disebut barisan $V$-ko-terpisah. Lemma Snake sering digunakan dalam keeksakan barisan sehingga perlu diselidiki eksistensinya dalam barisan quasi-eksak dan dualnya. Dalam tesis ini dibahas sifat-sifat barisan quasi-eksak dan dualnya, sifat-sifat barisan $U$-terpisah dan $V$-ko-terpisah, kemudian Lemma \textit{Snake} pada diagram komutatif yang barisnya merupakan barisan quasi-eksak atau dualnya. Selain itu, dibahas pula Lemma Schanuel yang mengaitkan 2 barisan quasi-eksak yang memuat modul proyektif atau modul injektif.

A sequence $ A \xlongrightarrow{f} B \xlongrightarrow{g} C$ of $R$-modules is said to be $U$-exact (at $B$) if there exist a submodule $U$ of $C$ such that $Im(f) = g^{-1} (U)$. A sequence $ A \xlongrightarrow{f} B \xlongrightarrow{g} C$ of $R$-modules is said to be $V$-coexact (at $B$) if there exist a submodule $V$ of $A$ such that $f(V) = Ker(g)$. A $V$-coexact sequence is the dual notion of a $U$-exact sequence. A short $U$-exact sequence with pre-image of $U$ that is a direct summand of $B$ is called a $U$-split sequence, while a short $V$-coexact sequence with image of $V$ that is a direct summand of $B$ is called a $V$-cosplit sequence. The Snake Lemma is often used in exact sequences so it needs to be investigate its existence in quasi-exact sequences and its dual notion. In this thesis we discuss the properties of quasi-exact sequences and its dual notion, the properties of $U$-split and $V$-cosplit sequences, then Snake Lemma on commutative diagram whose rows are quasi-exact sequences or its dual notion. Moreover, we also discuss Schanuel's Lemma which connect two quasi-exact sequences that contain the projective module or injective modules.

Kata Kunci : exact sequences, Snake Lemma, Schanuel's Lemma, quasi-exact, $U$-exact sequences, $V$-coexact sequences.