U.S. Defense Secretary Ashton Carter assured South Korea of his country's "ironclad" commitment to defending South Korea from North Korea's nuclear threats, the South Korean defense minister said Thursday following the communist country's nuclear test a day earlier.

"Secretary Carter reaffirmed the U.S.' ironclad defense commitment to South Korea, and this includes all kinds of extended deterrence assets," Defense Minister Han Min-koo said in a joint press release with Carter.

The two defense chiefs discussed over the phone on Wednesday how to tighten their security collaboration after the North conducted another nuclear test, its fourth.

Han and Carter also shared their determination that North Korea will not be recognized in the international community as a country that legitimately possesses nuclear capabilities, according to the joint release.

"Both ministers agreed that North Korea should pay a price that is proportional to the provocation," Han said.

South Korea and the U.S. are on the same page in condemning the latest North Korean nuclear test as "a clear violation of the international law" and "unpardonable provocations that pose threats to the peace and stability of the Korean Peninsula and the Asia-Pacific region," Han also noted.

The ministers will "tightly collaborate" under the South Korea-U.S. alliance to deal with threats from North Korea's nuclear arms, weapons of mass destruction and ballistic missiles, he added.

On its part, South Korea will push for the early installation of it Kill Chain and Korean Air and Missile Defense to guard against North Korea's missile and nuclear threats, Han said, referring to the indigenous defense systems, which Seoul had originally aimed to complete by the mid-2020s.

The allies' defense ministries will continue with their regular joint war exercises and devise customized deterrence systems, said the joint press release, which was read by Defense Minister Han, flanked by U.S. Forces Korea commander Gen. Curtis Scaparrotti and Joint Chiefs of Staff Chairman Lee Sun-jin on each side. (Yonhap)