Make sure you find out the original title. "Soukoku" is not used in daily language, it became more of a literary term. Rivalry itself can be translated as follows (if you use IE make sure to reload the page with Japanese encodings turned on):
rivalry
【@】ライバルリ-【変化】《複》rivalries
【レベル】6
【名】 競争{きょうそう}、競争意識{きょうそう いしき}、張り合うこと、対抗{たいこう}、対立関係
rivalry against
~と張り合うこと
rivalry among reporters
報道合戦{ほうどう がっせん}
rivalry between factions
派閥争い
rivalry between the two powers
2大国間の競争{きょうそう}
rivalry in a competitive entrance exam to enter a prestigious kindergarten
有名幼稚園入学{ゆうめい ようちえん にゅうがく}をめぐってのお受験競争{じゅけん きょうそう}
rivalry in a competitive entrance examination to enter a prestigious kindergarten
有名幼稚園入学{ゆうめい ようちえん にゅうがく}をめぐってのお受験競争{じゅけん きょうそう}
rivalry in love
恋の鞘当て{さや� て}
rivalry over territory
縄張り争い
Rivalry with best friend // Rivalry with one's best friend
親友との競争
rivalry within the Church
教会内{きょうかい ない}の抗争{こうそう}
Besides, if you use the word "soukoku" in katakana, it is difficult to grasp the word's meaning.