“I’m running for president,” Mr. Cruz said in a Twitter post, becoming the first major candidate of either party to enter the race and heightening his national visibility.

The announcement by Mr. Cruz marks the beginning of the primary election battle to define a Republican Party that is divided about the balance between ideology and pragmatism, and which is uncertain about who should lead it. His candidacy comes as recent polls indicate that none of the likely candidates has yet emerged as a bridge-building consensus choice among the party’s factions.

Mr. Cruz will be planting his flag on the far right flank of what is expected to become a crowded primary field spanning an ideological spectrum from centrists, such as New Jersey Gov. Chris Christie, to the libertarian Kentucky Sen. Rand Paul and social conservatives like Mike Huckabee, a former governor of Arkansas.

Mr. Cruz stands for a brand of ideological conservatism that contrasts with former Florida Gov. Jeb Bush, who has said a successful presidential candidate must be willing to “lose the primary”—that is, risk angering the party’s most conservative followers—to succeed with the more centrist electorate in the general election.

By contrast, Mr. Cruz has sometimes angered congressional leaders by pushing for conservative goals, such as ending the Democratic-backed Affordable Care Act and President Barack Obama’s immigration policy, at the cost of provoking gridlock in Washington. He was widely blamed for helping to prompt the 2013 government shutdown, an effort to unwind the 2010 health law that many Republicans say wound up damaging the party’s image.

Mr. Cruz is scheduled to speak Monday at Liberty University, a Christian college in Lynchburg, Va., founded by the Rev. Jerry Falwell.

Mr. Cruz’s early presence in the field could pressure other potential candidates to move to the political right on fiscal policy, social issues and on the tactics for pursuing policy goals, particularly among contenders who are trying to establish themselves as the leading conservative alternative to Mr. Bush.

“Cruz has the potential to take up quite a bit of space on the right in the conservative primary-within-the-primary,” said Kevin Madden, a veteran GOP campaign strategist. “The candidates looking to grab the mantle of being the antiestablishment choice for voters will certainly start to feel pressure to match Cruz step for step and not allow themselves to get outflanked to their right.’’

Mr. Cruz argues that the GOP repeatedly has lost the White House because it has rejected strict conservatives in favor of more centrist candidates.

“If we run another candidate in the mold of a Bob Dole or a John McCain or Mitt Romney, we will end up with the same result, which is millions of people will stay home on Election Day…If we run another candidate like that, Hillary Clinton will be the next president,” Mr. Cruz said in a 2014 CNBC interview.

By jumping in first for the most wide-open GOP nomination fight in a generation, Mr. Cruz is hoping to claim a measure of visibility. Other potential candidates are preparing to formally declare their bids in the coming weeks.

Well known public figures like Messrs. Bush and Christie seem to be in less of a rush to launch races and inaugurate a more intense stage of campaigning and scrutiny.

Still in his first term—he was elected in 2012—Mr. Cruz has used his place in the Senate to define himself as one of the most ardent conservatives and as more aggressive than party leaders in fighting for policy goals. One example came in 2013, when he delivered a 21-hour speech on the Senate floor to dramatize his opposition to funding for the health-care law.

He highlighted that contrast in a recent video he called “Truth,” which implied that other Republican candidates offer more talk than action.

“Obamacare: When have you stood up and fought against it?” Mr. Cruz said in the video. “President Obama’s illegal and unconstitutional executive amnesty: When have you stood up and fought against it?” he asked, referring to the president’s actions, without congressional approval, to shelter many illegal immigrants from deportation.

Mr. Cruz’s supporters contend he is in a good position to appeal to voters across party lines who are disillusioned with the political system. “There is clearly a frustration in this country with the status quo, and people are looking for something new and something different,” said Saul Anuzis, former chairman of the Michigan Republican Party.

But Mr. Cruz’s Republican critics say he is too polarizing to be a strong general election candidate.

“He thinks he has a formula for energizing conservatives, but he doesn’t have the skill or inclination to reach out to other people,” said Pete Wehner, a Republican strategist who worked in the George W. Bush White House. “I don’t think he’s got much appeal beyond the core base of the Republican Party.”

A Wall Street Journal/NBC News poll this month suggested that Mr. Cruz had work to do to boost himself among Republican primary voters. Some 40% said they could see themselves backing him for president, while 38% said they couldn’t.

Democrats already are criticizing Mr. Cruz as someone outside the mainstream on issues including climate change, which Mr. Cruz has said isn't supported by science.

“That man betokens such a level of ignorance and a direct falsification of the existing scientific data…[he] has rendered himself absolutely unfit to be running for office,” said California Gov. Jerry Brown, speaking Sunday on NBC.

(Published by The Wall Street Journal - March 23, 2015)