CAPE GIRARDEAU, Mo.—Rex Dearmont Rust, co-president of the Missouri-based media company Rust Communications, has died. He was 52. The company’s flagship newspaper, the Southeast Missourian in Cape Girardeau, reported that Rust died Thursday evening after a yearlong battle with cancer. Rust grew up in Cape Girardeau and graduated from Harvard University in 1992. He worked in finance in New York, Chicago, and Virginia before returning to Missouri. In 2001, Rust and his brother, Jon K. Rust, were named co-presidents of the company founded by their father, Gary Rust, who continued on as chairman of the board. Today, Rust Communications owns more than 40 newspapers in eight states, numerous magazines and websites, and has minority ownership of 17 radio stations. Friends and family noted Rust’s community service. He was a member of several community and professional organizations. U.S. District Judge Stephen Limbaugh Jr. recalled once inviting Rust to play golf. He …